ff    r  <::^^ 


.  ^.  I^ha^oras  '(aboU't  569-500  bc.)  settled  in  Croton,  a  Dorian 
•  •  (fcv^fiy^jVi'36utlteiriT'^  Italy,  where  he  opened  a  school  in  which 
philosophy  and  mathenriatics  were  taught.  He  founded  a  brother- 
hood, the  members  of  which  were  afterwards  called  Pythagoreans. 
He  is  said  to  have  taught  that  the  foundation  of  the  theory  of  the 
universe  was  to  be  found  in  the  science  of  numbers.  The  word 
mathematics  has  been  ascribed  to  him. 


ELEMENTARY  ALGEBRA 


BY  . 


GEORGE  H.   HALLETT,  A.M.,  Ph.D. 

PROFESSOR   OF   MATHEMATICS 
THE   UNIVERSITY   OF  PENNSYLVANIA 

AND 

EGBERT  F.   ANDERSON,   A.M.,   Sc.D. 

PROFESSOR   OF  MATHEMATICS 
STATE  NORMAL  SCHOOL,   WEST   CHESTER,   PA. 


SILVER,   BURDETT  AND   COMPANY 

BOSTON  NEW  YORK  CHICAGO 


aAi54 


Copyright,  1917, 
By  silver,  BURDETT  AND  COMPANY 


SDUCATION  DEFT^ 


Ol^ 


PREFACE 

This  book  is  designed  primarily  for  the  use  of  those  who 
are  beginning  the  study  of  algebra ;  it  is,  however,  sufficiently 
extensive  to  serve  as  a  text  for  a  review  of  algebra  during  the 
third  or  fourth  year  of  the  high  school  course  in  those  schools 
in  which  the  curricula  call  for  such  a  review. 

In  preparing  this  text  the  authors  have  kept  in  view  the 
fact  that  no  substantial  progress  in  algebra  is  possible  for  the 
student  unless  due  emphasis  is  placed  on  the  fundamental  prin- 
ciples and  processes  of  the  subject,  and  that  these  essentials 
must  be  provided  for  irrespective  of  whatever  trend  the  teach- 
ing of  algebra  may  have  had  within  the  last  few  years,  or  may 
have  in  the  future. 

Long  experience  in  teaching  has  convinced  the  authors  that 
the  technical  terms  employed  and  the  principles  involved  in 
problems  of  physics  and  engineering  are  not  sufficiently  under- 
stood to  warrant  their  inclusion  in  an  elementary  course  in 
algebra  ;  that  practically  the  whole  of  such  a  course  should  be 
devoted  to  the  treatment  of  the  elements  of  the  subject  itself, 
and  that  the  problems  referred  to  may  be.  solved  with  little 
difficulty  by  the  student  who  has  mastered  the  first  course  in 
algebra  before  he  begins  the  study  of  physics.  However,  they 
believe  that  formulae  drawn  from  various  sources,  including 
physics  and  engineering,  should  be  used  extensively  in  ele- 
mentary algebra;  for  there  is,  perhaps,  no  more  important 
practical  exercise  in  the  subject  than  that  which  comes  from 
determining  the  actual  or  approximate  numerical  value  of  a 
literal  number  which  occurs  in  a  formula,  when  given  numerical 
values  are  substituted  for  the  remaining  letters  of  the  formula. 

To  make  effective  provision  for  attaining  the  end  in  view, 

54  5  -jftU 


iv  PREFACE 

namely,  the  furnishing  of  a  textbook  from  which  the  student 
may  acquire  a  thorough  grounding  in  the  elements  of  algebra, 
the  authors  have  made  its  prominent  features  the  following : 

1.  The  simplest  possible  presentation  of  the  topics  of  elemen- 

tary algebra. 

2.  The  use  of  the  inductive  method  in  developing  fundamental 

concepts  and  principles. 

3.  The  use  of  illustrative  problems  to  make  clear  the  applica- 

tion of  the  principles. 

4.  The  actual  application  of  the  principles  by  means  of  nu- 

merous, well-graded  examples,  both  sight  and  written. 

5.  The  extensive  use  of  numerical  checks  to  promote  habits 

of  accuracy  and  to  give  the  student  confidence  in  the 
results  of  his  work. 

6.  The  copious  supply  of  problems  designed  to  give  the  stu- 

dent facility  in  applying  the  mechanics  of  algebra. 

7.  The  reviews  designed   to  make   the  students'  knowledge 

cumulative  and  coherent  and  thorough. 

8.  Such  treatment  of  graphs  as  is  necessary  to  render  the 

student  familiar  with  the  underlying  principles  of 
graphical  representation  so  that  he  may  be  able  to 
apply  them  wherever  there  may  be  need  for  their 
application. 

9.  The  conciseness  and  exactness  of  statement  of  definitions 

and  principles. 

The  author  wishes  to  gratefully  acknowledge  the  courtesy 
of  the  Open  Court  Publishing  Company  in  permitting  the  re- 
production of  the  portraits  contained  in  this  book. 


CONTENTS 

OHAPTEB  PAGE 

I.  Introduction 1 

n.  Fundamental  Processes 34 

m.  Simple  Equations 81 

rv.  Type  Products  and  Factors      .        .        .        .        .  100 

V.  Fractions 153 

VI.  Fractional  and  Literal  Equations         .        .        .  191 

Vn.  Systems  of  Linear  Equations 205 

Vni.  Katio,  Proportion,  and  Variation    .        ...        .  239 

IX.  Graphs 251 

X.  Powers,  Roots,  Radicals,  and  Exponents       .        .  262 

XI.  Involution  and  Evolution 300 

XII.  Quadratic  Equations 312 

XIII.  Systems  of  Quadratic  Equations     ....  350 

XIV.  Progressions 365 

XV.  General  Review 381 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/elementaryalgebrOOhallrich 


ELEMEI^TARY  ALGEBRA 


CHAPTER   I 

INTRODUCTION 

The  Notation  of  Algebra 

1.  The  Notation  of  Arithmetic.  In  arithmetic  numbers 
are  expressed  by  the  symbols  0,  1,  2,  3,  4,  5,  6,  7,  8,  9. 
The  numbers  represented  by  these  symbols  are  called 
integers.  The  operations  of  addition,  subtraction,  multi- 
plication, and  division  performed  on  these  integers  lead 
either  to  integers  or  to  fractions.  Therefore,  primarily, 
arithmetic  treats  of  integers  and  fractions  and  the  opera- 
tions which  are  indicated  by  the  signs  +,  — ,  X,  and  -?-. 

2.  Algebra.  Algebra,  like  arithmetic,  treats  of  number. 
The  symbols  which  are  used  in  arithmetic  are  retained  in 
algebra.  In  algebra,  however,  new  symbols  of  number 
and  of  relations  between  numbers  are  introduced,  and,  in 
the  written  language  of  algebra,  systematic  use  is  made 
of  letters  to  represent  numbers.  A  number  which  is  repre- 
sented by  a  letter  is  called  a  literal  number. 

3.  Use  of  literal  numbers.     In  arithmetic  it  is  shown 

2     4      2x4 
that  -x-  =  - -•     This  example  illustrates  a  principle 

d        O        O  X  O 

which  may  be  expressed  in  words  as  follows  : 

The  product  of  two  fractions  is  equal  to  a  fraction 
whose  numerator  is  the  product  of  the  two  given  numerators 

1 


2  ELEMENTARY  ALGEBRA 

and  whose  denominator  is  the  product  of  the  two  given 
denominators. 

This  statement  may  be  expressed  concisely  in  the  writ- 
ten language  of  algebra  thus : 

a^c_ axe 
b     d     b  X  d 

Here  a  denotes  the  integer  which  is  the  numerator  of  the  first  frac- 
tion and  b  the  integer  which  is  its  denominator ;  also,  c  denotes  the 
integer  which  is  the  numerator  of  the  second  fraction  and  d  that 

which  is  its  denominator.     In  the  algebraic  expression  -,  the  letters 

a  and  b  represent  any  integers  whatever,  whereas  in  the  arithmetical 
expression  l^,  the  symbols  2  and  3  denote  definite  integers. 

4.  Symbols.  The  language  of  algebra  employs  symbols 
to  represent:  (1)  Numbers  themselves;  (2)  operations 
on  numbers;   (3)  relations  between  numbers. 

5.  Symbols  of  number.  Numbers  in  algebra  are  ex- 
pressed by  Arabic  numerals,  and  also  by  letters  ;  thus; 

c  may  represent  the  number  of  cents  in  the  cost  of  an  orange ; 
a  dollars  may  denote  any  number  of  dollars ; 
m  may  stand  for  the  number  of  miles  between  two  towns ; 
X  may  stand  for  any  number,  which  for  the  sake  of  brevity  is  called 
the  number  x. 

6.  Symbols  of  operation.  Symbols  of  operation  are  as 
follows : 

-}-  is  the  sign  of  addition  ;  it  is  read  plus. 
7  +  3  denotes  the  sum  of  the  two  numbers  7  and  S;  a  -\-  h  stands 
for  the  sum  of  the  two  numbers  represented  by  the  letters  a  and  b. 

—  is  the  sign  of  subtraction  ;  it  is  read  minus. 

a  —  b  stands  for  the  difference  of  the  numbers  represented  by  the 
letters  a  and  b. 

X  is  the  sign  of  multiplication;  it  is  read  times  or  mul- 
tiplied hy» 


INTRODUCTION  3 

a  y.  h  stands  for  the  product  of  the  numbers  represented  by  the 
letters  a  and  b.  Multiplication  is  also  indicated  by  a  dot.  2  x  a 
may  be  written  2  •  a.  When  numbers  are  represented  by  letters,  the 
sign  of  multiplication  may  be,  and  usually  is,  omitted. 

Thus,  5  ab  stands  for  5  x  a  x  &. 

-f-  or  :  is  the  sign  of  division;  it  is  read  divided  hy, 

3  -f-  5  or  3  :  5  denotes  the  quotient  obtained  from  the  division  of 
3  by  5. 

Remark.     Neither  the  sign  -h  nor  the  sign  :  is  so  frequently  used 

in  algebra  as  formerly.     For  instance,  a  -4-  &  is  usually  written  -. 

b 

7.  Symbols  of  relation.  Symbols  of  relation  are  as 
follows  : 

=  is  the  sign  of  equality  and  is  read  equals^  or  is  equal  to. 

a  =  b  expresses  the  equality  of  the  numbers  represented  by  the 
letters  a  and  b. 

>  and  <  are  signs  of  inequality.  >  is  read  greater 
than  ;   <  is  read  le%%  than. 

a'>b  means  that  the  number  represented  by  a  is  greater  than  the 
number  represented  by  6.  a  <  6  means  that  the  number  represented 
by  a  is  less  than  the  number  represented  by  b. 


EXERCISE  1 

1.  In  the  following,  c  stands  for  cost  (meaning,  say, 
the  number  of  cents  in  the  cost),  s  for  selling  price,  g  for 
gain,  and  I  for  loss  ;  read  in  words  : 

c-{-g  =  8.  8-hl=o.  8—c  =  g. 

c—  8  =  1.  8  —  g  =  e,  c—  I  =  8. 

2.  3  +  2  expresses  the  sum  of  3  and  2 ;  what  then  is 
expressed  by  3  +  5?     5  + a?     5  +  6?     a+b?  t 

3.  6  —  2  expresses  the  difference  of  6  and  2 ;  what  then 
is  expressed  by  8-5?     7-a?     6-3?     a-b? 


4  ELEMENTARY  ALGEBRA 

4.  3x5  expresses  the  product  of  3  and  5  ;  what  then 
is  expressed  by  4x6?     3xa?     4:  x  b?    a  xh? 

5.  What  is  expressed  by3.c?     a  »  b?     I  x  w?     ab? 

f*  ♦ 

6.  -  expresses  the  quotient  of  6  divided  by  2  ;  what 

then  is  expressed  by  ??     ^?     h     ^?     £?     4^?     ^? 
^  '^42a2cJ5r^ 

7.  Using  the  +  sign,  express  the  sum  of  6  and  7  ;  5 
and  a  ;  c  and  d. 

8.  Using  the   —   sign,  express  the  difference  of  7  and 

6  ;  a  and  7  ;  3  and  b  ;  a  and  b. 

9.  Using  the  x  sign,  express  the  product  of  2  and  5 ; 

7  and  a  ;  3  and  b  ;  a  and  6. 

10.  Indicate  in  three  ways  the  product  of  3  and  a  ;  4 
and  b  ;  c?  and  c?. 

11.  Using  the  fractional  form,  indicate  the  quotient  of 
9  divided  by  4  ;  a  divided  by  3  ;  6  divided  by  2  ;  a 
divided  by  c. 

12.  Find  the  number  that  is  equal  to  a  + 1  when  a 
stands  in  turn  for  3 ;  5  ;  7  ;  12  ;  J  ;  1|  ;  0  ;  .3. 

13.  Find  the  number  that  is  equal  to  b  —  1  when  b 
stands  in  turn  for  2  ;  3;  7;  IJ  ;  1;  2^;  1.5;  2.25. 

14.  Find  the  number  that  is  equal  to  2  a  when  a  stands 
in  turn  for  2  ;  3 ;  5  ;  |  ;  1 J  ;  .5  ;  1.1. 

15.  Find  the  number  that  is  equal  to  ^  when  a  stands 
in  turn  for  2  ;  1 ;  3  ;  6  ;  J  ;  5  ;  IJ. 

16.  When  a  stands  for  2  and  b  stands  for  1,  what  num- 
ber is  equal  to  a  +  6  ?     a  —  b?     U'b?     -? 

0 


INTRODUCTION  5 

17.  When  x=  6  and  ?/  =  3,  what  number  is  equal  to 

1/1/  2.y 

18.  When  a;  =  |^  and  i/  =  ^i  what  number  is  equal  to 
x-\-i/?     x-yl     x-yl     -?     — ^? ? 

y       y       ^-y 

19.  If  goods  cost  c  dollars  and  were  sold  at  a  gain  of  g 
dollars,  express  the  selling  price.  If  c  =  7  and  ^  =  3,  what 
was  the  selling  price  ? 

20.  If  goods  cost  p  dollars  and  were  sold  at  a  loss  of 
t  dollars,  express  the  selling  price.  If  jo  =  9  and  <  =  2, 
what  was  the  selling  price  ? 

21.  Express  the  number  that  is  6  more  than  a. 

22.  Express  the  number  that  is  c  more  than  d. 

23.  Express  the  number  that  is  5  less  than  c. 

24.  Express  the  number  that  is  n  less  than  m. 

25.  Express  the  sum  of  a,  6,  and  c. 

26.  Express  the  sum  of  a  and  h  diminished  by  c. 

27.  Find  the  cost  of  3  pounds  of  sugar  at  a  cents  a 
pound. 

28.  Find  the  cost  of  a  yards  of  cloth  at  h  cents  a  yard. 

29.  If  p  pounds  of  sugar  cost  30  cents,  what  did  one 
pound  cost  ? 

30.  If  h  bushels  of  grain  weigh  p  pounds,  what  does 
one  bushel  weigh  ? 

31.  What  did  a  boy  pay  for  2  oranges  at  c  cents  each 
and  3  apples  at  h  cents  each  ? 

32.  How  much  change  should  a  boy  receive  from  c 
cents  given  in  payment  for  a  ball  that  cost  h  cents? 

33.  Express  the  result  of  adding  p  times  r  to  I. 


6  ELEMENTARY  ALGEBRA 

34.  Express  the  result  of  adding  t  times  r  to  p. 

35.  Express  the  result  of  dividing  Vhy  a  times  h. 

36.  Write  the  product  of  3,  a,  6,  and  c  in  three  different 
ways. 

37.  How  many  quarts  are  there  in  a  gallons  ? 

38.  How  man^  quarts  are  there  in  h  pecks  ? 

39.  How  many  pecks  are  there  in  c  bushels  ? 

40.  How  many  quarts  are  there  in  m  pints  ? 

41.  How  many  units  are  there  in  c  dozen  ? 

42.  How  many  dozen  are  there  in  u  units  ? 

43.  If  I  stands  for  the  length  of  one  line  in  inches  and 
w  for  the  length  of  another  line  in  feet,  what  stands  for 
the  sum  of  their  lengths  in  inches  ?  What  stands  for  the 
difference  of  their  lengths  in  feet  ? 

44.  If  I  stands  for  the  number  of  inches  in  the  length 
of  a  rectangle  and  w  for  the  width  in  inches,  what  stands 
for  the  number  of  inches  in  the  perimeter  ? 

45.  If  8  stands  for  the  length  of  one  side  of  an  equi- 
lateral triangle,  what  stands  for  the  perimeter  ? 

8.  Algebraic  expressions.  Any  symbol  or  combination 
of  symbols  used  in  algebra  to  express  a  number  is  called 
an  algebraic  expression,  or  simply  an  expression. 

Thus,  2  a,  a  +  &,  — ,  3  ar  —  y  +  «  are  algebraic  expressions. 

9.  Evaluation  of  an  algebraic  expression.  The  process 
of  substituting  numbers  for  letters  in  an  expression  and 
calculating  tlie  numerical  value  of  the  result  is  called  the 
evaluation  of  the  expression  for  the  given  values  of  the 
letters. 


INTRODUCTION  7 

Thus,  to  evaluate  2  a  -\-  b  when  a  =  2  and  6  =  3,  substitute  the  given 
values  of  a  and  b  in  the  expression,  obtaining  2-2  +  3=4  +  3  =  7. 

Remark.  It  is  often  convenient  in  testing  the  accuracy  of  alge- 
braic work  to  evaluate  the  expression  for  simple  numerical  values  of 
the  letters.     This  is  termed  checking  the  result. 

10.  Order  of  operations.  Algebraic  expressions  often 
contain  different  signs  of  operation.  In  evaluating  such 
expressions  it  is  understood  that : 

When  operations  of  addition  and  subtraction  are  indicated 
in  an  expression^  they  are  to  he  performed  in  the  order  of 
their  occurrence  from  left  to  right. 

Thus,  4  +  5-2  +  3-6  =  9-2  +  3-6  =  7+3-6  =  10 -6=  4. 

When  operations  of  multiplication  and  division  are  indi- 
cated in  an  expression^  they  are  to  he  performed  in  their 
order  from  left  to  right  and  before  the  operations  of  addition 
and  subtraction  are  performed. 

Thus,  6  X  8-^4  +  12 -6  X2-24 -4 -2=48-4  +  2  x2- 6-^2 

=  12  +  4-3  =  13. 

EXERCISE  2 

1.  What  is  the  value  of  2  a  —  6  when  a  =  l  and  5  =  1? 

2.  What   number   is  when    a  =  2   and    5  =  4? 

2 

When  a  =  1  and  5  =  3  ?     When  a  =  J  and  5  =  J  ? 

3.  If  a  stands  for  the  number  of  units  in  the  altitude, 
h  for  the  number  in  the  base,  and  A  for  the  area  of  a 
rectangle,  read  in  words  the  statement,  A  =  ah. 

4.  What  is  the  area  of  a  rectangle  when  the  altitude 
equals  5  ft.  and  the  base  equals  7  ft.? 

5.  What  is  the  area  of  a  rectangle  when  a  =  3  and 
6  =  4? 


8  ELEMENTARY  ALGEBRA 

6.  A  man  walked  3  hours  at  the  rate  of  4  miles  an  hour. 
How  far  did  he  walk  ?  If  he  walked  x  hours  at  the  same 
rate,  how  far  did  he  walk  ? 

7.  A  man  walked  a  hours  at  the  rate  of  x  miles  an 
hour.     How  far  did  he  walk  ? 

8.  What  is  the  cost  of  5  yards  of  cloth  at  x  cents  a 
yard? 

9.  How  many  a's  in  a  + a?  In2a4-a?  In3a-fa? 
In3a  +  2a?     Tn4a-a?     In  5a- 2a?     In7a-3a? 

In  examples  10-16,  supply  the  missing  numbers. 

10.  4x  $10  +  2  X  110  +  3  X  $10  =  (     )x$10. 

11.  4xa  +  2xa  +  3xa=(     )xa. 

12.  6  X  10  ft.  +  5  X  10  ft.  -  3  X  10  ft.  =  (     )  X  10  ft. 

13.  6x6  +  5x6-3x6=  (     )x6. 

14.  6a;  +  5a;  +  4a;  —  2aj=(     )a:. 

15.  7w+5m  —  3m— 2m=(     )w. 

16.  9r  +  5r— 3r-r=(     )r. 

17.  A  boy  is  a  years  old,  his  father  is  2  a  years  old. 
How  many  years  are  there  in  the  sum  of  their  ages  ? 

18.  A  person  is  x  years  old.  How  old  will  he  be  two 
years  hence?  How  old  was  he  2  years  ago?  How  old 
was  he  a  years  ago  ? 

19.  One  number  is  three  times  another.  If  n  represents 
the  smaller  number,  what  expression  represents  their  sum  ? 
Their  difference  ? 

20.  Some  sugar  costing  a  dollars  was  sold  at  a  gain  of 
50%.     State  the  selling  price. 

21.  The  circumference  of  a  circle  is  equal  to  2  tt  times 
the  radius;  this  truth  may  be  briefly  expressed  thus: 

c  =  2  Trr. 


INTRODUCTION  9 

Find  the  length  of  the  circumference  if  7r  =  ^  and  r  =  7; 
if  7r=^andr  =  |. 

22.  The  radius  of  a  circle  is  x  ft.     What  is  the  circum- 
ference ? 

23.  How  much  greater  than  a  is  a  +  2  ?     How  much 
greater  than  a  -{- 1  is  a  -^  2? 

24.  Read  the  expression  2a  -  a;  also,  a  *  2  a. 

Assume   that   a  =  5,  5  =  2,   c  =  1,   a;  =  3,   y  =  2,   2  =  0, 
and  evaluate  the  following : 


25. 


y  x-z 

4:  ab-\-  ac—bc 


27.    2  a5tf  —  X1/Z,  28. 

29.      ^^     I      ^^^     I     25^ 


2ic^ 


30. 


a  +  b      2a;+l      2^-f-a; 
2a. 35       x  +  l        20y 
a4-5  +  <?        4«/        3a;— 1 


11.  Laws  of  combination  in  algebra.  The  laws  of  com- 
bination of  algebraic  expressions  are  essentially  the  same 
as  the  laws  of  combination  of  numbers  in  arithmetic;  for, 
in  general,  by  substituting  numbers  for  the  letters,  the 
algebraic  expressions  become  arithmetical.  Constant  ref- 
erence to  this  principle  will  be  made  in  developing  the 
fundamental  laws  of  algebra. 

12.  Factors.  Each  of  the  numbers  whose  product  is  a 
given  number  is  called  a  factor  of  the  given  number. 

Thus,  since  12  =  2  x  6,  each  of  the  numbers  2  and  6  is  a  factor  of 
12.  Since  2xy  =  2  >  x  >  y,  each  of  the  numbers,  2,  x,  and  y,  is  a  factor  of 
2xy. 

Remark  1.  The  expression  2(x  +  y)  means  2  times  the  sum  of  x 
and^. 


10  ELEMENTARY  ALGEBRA 

Thus,  the  factors  of  2(-c  +  y')  are  2  and  (x  +  y). 
Remark  2.    A  number  may  have  different  sets  of  factors. 
Thus,  sets  of  factors  of  24  are  4  and  6,  2  and  12,  3  and  8 ;  2,  2,  and 
6 ;  2,  3,  and  4 ;  and  2,  2,  2,  and  3. 

13.  Coefficient.  When  a  number  is  the  product  of  two 
factors,  either  of  these  factors  is  called  the  coefficient  of 
the  other  in  the  product. 

.  Thus,  in  5  ah,  5  is  the  coefficient  of  ah,  and  5  a  is  the  coefficient  of  h. 

Note  1.    A  numerical  coefficient  is  usually  written  first. 

Thus,  the  product  of  2  and  a  is  written  2  a,  not  a  2. 

Note  2.    The  numerical  coefficient  1  is  usually  omitted. 

Thus,  1  X  a  is  usually  written  a. 

Remark.  When  the  term  coefficient  is  used,  numerical  coefficient 
is  usually  meant. 

Thus,  in  3  a&,  3  is  understood  to  be  the  coefficient,  unless  otherwise 
implied. 

EXERCISE  3 

1.  Give  factors  of  18  ;  14  ;  96  ;  23. 

2.  Omitting  the  factors  1  and  3  a6,  name  six  factors  of 

3.  Name  the  factors  of  3(a  -|-  5). 

4.  Name  three  sets  of  factors  of  abc. 

5.  What  is  the  coefficient  of  ah  in  3  ai  ? 

6.  What  is  the  coefficient  oi  2  xy  m  ^  ah  -  '^^xyl 

7.  Each  factor  in  2xyz  has  a  coefficient.     Name  each 
factor  and  give  its  coefficient. 

8.  Write  the  factors  of  a(h  +  c);  2  a(h-{-  c). 

9.  Write  the  product  a  x  2  x  6  in  its  usual  form. 

10.  Write  1  .  a  4- 1  •  5  in  a  better  form. 

11.  Is  the  following  a  true  statement:  l.a+2.a 
=  (H.2)a? 


INTRODUCTION  11 

14.  Powers.  The  product  of  two  equal  factors  is  called 
the  square,  or  second  power,  of  one  of  the  factors.  Similarly, 
the  product  of  three  equal  factors  is  called  the  cube,  or 
third  power,  of  one  of  the  factors;  and  the  product  of  four 
equal  factors  is  called  the  fourth  power,  etc. 

Thus,  2  X  2  is  the  square,  or  second  power,  of  2 ;  a  x  a  is  the 
square  of  a;  aaa  is  the  third  power  of  a;  and  aaaaaa  is  the  sixthi 
power  of  a. 

15.  Base  and  exponents.  For  convenience,  a  x  a  is 
usually  written  a^,  read  a  square ;  a  x  a  x  a  is  written  a^ 
read  a  cube ;  ax  a  x  a  x  a  is  written  a*,  read  a  to  the 
fourth  power  or  a  with  an  exponent  Jf.  In  an  expression 
similar  to  a*,  the  number  a  is  called  the  base,  and  the 
number  4  is  called  the  exponent,  or  index.  When  the  ex- 
ponent is  an  integer  it  shows  how  many  times  the  base  is 
used  as  a  factor. 

Thus,  in  2^  =  64,  the  number  64  is  the  sixth  power  of  2,  and  the 
exponent.,  6,  shows  that  the  ftcwe,  2,  is  used  six  times  as  a  factor  in  ob- 
taining 64. 

Note.  The  exponent  1  is  usually  omitted.  Thus,  a}-  is  usually 
written  a. 

Remark  1.  In  a  subsequent  chapter  other  numbers  than  integers 
will  be  introduced  as  exponents.     Their  use  will  then  be  explained. 

Remark  2.  Care  should  be  taken  not  to  confuse  an  exponent 
with  a  coefficient. 

Thus,  3  a  means  a  +  a  +  «,  while  a^  means  a  x  a  x  a. 

Remark  3.  It  should  be  emphasized  that  such  expressions  as 
2  ah^  mean  2  dbh  and  not*  2ab  x2  ah.  The  latter  expression  is  written 
(2  ahy. 

EXERCISE  4 

Write  examples  1-5  in  another  form,  using  exponents 
and  coefficients : 

1.  2.2.2.2.2. 

2.  a  +  a  +  a-f-a  +  a. 


12  ELEMENTARY  ALGEBRA 

3.  aa  4-  CL(X" 

4.  XXX  -\-  XX  -{-  XX  +  1, 

5.  aaa  +  aaa  —  aa -{•  a  +  a -\-  a. 

6.  Evaluate  each  of  the  following  when  x  =  2: 

a^;  ar^;  a:*;  a^. 

7.  Evaluate  each  of  the  following  when  a;  =  3 : 

2x^;   bx^;  ^x^;   2  3^;   ^  x^. 

8.  Evaluate  a^  +  2x^  —  x-\-2  when  a;  =  3. 

9.  Evaluate  a^—  b^  +  2  ab  when  a  =  3  and  6  =  2. 

10.  Evaluate  a^  —  h^  when  a  =  2  and  6  =  1. 

11.  Evaluate  3  a^  4.  2  6  —  5  when  a  =  2  and  6=1. 

12.  Evaluate  y^—2y-{-l  when  y  =  2. 

Rewrite  these  expressions,  using  exponents  when  possible : 

13.  2  a6  X  2  a6  X  2  ah. 

14.  3  a26  X  3  a^h  x  3  a%. 

15.  3  aa  -  2  66 ;  aaa  -  2  x  3  66 ;  2  •  2  .  2  aa  -  3  x  3  66. 

16.  2  x2aa-3  X  3666;  2  x  2  x  2a;a;  -  2  x  3yy ;  25aa 
-  49  66. 

16.  Parentheses.  Parentheses,  (  ),  are  used  to  indi- 
cate that  the  expression  inclosed  is  to  be  treated  as  a  single 
number.  When  different  parentheses  are  used  in  the  same 
expression,  other  forms,  as  brackets,  [  ],  braces,  |  j,  and 
the  vinculum, ,  are  employed. 

Thus,  a  +{1)  —  c)  means  that  the  number  remaining  after  subtract- 
ing c  from  h  is  to  be  added  to  a.     (5  +  4)(6  —  2)  means  the  product 

of  9  and  4.    In  — -^,  the  vinculum  groups  a  +  6  into  one  number,  and 

c  +  a 
c  -\-  d  into  one  number. 

Note.     The  various  forms   of   parentheses   are  called  signs  of 
aggregation ;  their  use  is  illustrated  by  the  following  expression  : 
[2  +  {3  -  2}  +  3^  +(2  +  5)  +  (4  -  2)]  =  [2  +  1  +  2  +  7  +  2]  =  14. 


INTRODUCTION  13 

EXERCISE  5 

Simplify : 

1.  (2  +  3)(4  +  5). 

Thus,  (2  +  3)(4  +  5)  =  5  X  9  =  45. 

2.  (5  +  3)(8-3).  ^       3.  5(3  +  4-5). 

4.  (18-12)^3.  5.  15-(3x2  +  4). 

6.  (12  +  6)H-(8-5).  7.  (15-5)-f-(3  +  2). 

8.  (2  4-4)(6-4)^(5-hl).  9.  (8 +  4) -5- (10 -6) (15 -5). 

^^    (12  +  3)x(12-7),         ^^    2  +  [3+(l  +  15)]. 

2x5 
12.  3  +  (2-l)  +  (3  +  2).        13.    8  +  [4-f.2  +  3]. 

14.  (3  +  l)(2  +  l)+2. 

15.  5[2  +  (3-2)  +  j3  +  2-|-in. 

16.  15^^  X  2  H- 8T6"-J- 2. 

17.  Monomial.  An  algebraic  expression  which  does 
not  contain  an  addition  or  a  subtraction  sign  is  called  a 
monomial  expression,  or  simply  a  monomial. 

Thus,  3  a,  —,   and      ^      are  monomials. 
b  d 

Note.     An  expression  within  parentheses  must  be  regarded  as  a 

monomial,  since  it  is  to  be  taken  as  a  whole. 

Thus,  2(a  +  b)  and  (a  -f-  a)  are  monomials ;  but  a  +  6  is  not  a 
monomial. 

18.  Terms.  The  monomials  of  an  algebraic  expression 
which  are  connected  by  -f  and  —  signs  are  called  the 
terms  of  the  expression. 

Thus,  the  terms  of  aa;  +  3  6  +  2  c  are  ax,  3  b,  and  2  c. 

19.  Binomial.  An  algebraic  expression  of  two  terms  is 
called  a  binomial. 

Thus,  a  +  2,  3  a  -  X,  and  2(a  -{-  b)  -\-  'd(x  +  y)  are  binomials. 


14  ELEMENTARY  ALGEBRA 

20.  TrinomiaL  An  algebraic  expression  of  three  terms 
is  called  a  trinomiaL 

Thus,  X  +  y  +  z,  2a  +  3b  -  c,  and  2(b  +  c)  +  3(a;  +  y)  +  c(a  +  b) 
are  trinomials. 

21.  PolynomiaL  An  algebraic  expression  of  two  or 
more  terms,  is,  in  general,  called  a  polynomial. 

Thus,  a  binomial  is  a  polynomial  of  two  terms,  and  a  trinomial  is 
a  polynomial  of  three  terms. 

22.  Like  terms.  Terms  which  do  not  differ  except  in 
their  coefficients  are  called  like  terms. 

Thus,  3ab%  6ab%  and  ^ab^  are  like  terms;  but  3ab^  and  5a%  are 
unlike  terms. 

Remark.  Terms  may  be  regarded  as  like  with  respect  to  a  cer- 
tain factor  or  certain  factors,  when  the  remaining  factors  are  regarded 
as  coefficients.  Thus,  ay  and  by  are  like  terms  with  respect  to  y ;  also 
abx  and  cbz  are  like  with  respect  to  bx. 

EXERCISE  6 

1.  How  many  terms  are  there  in  2x—  3^  +  «? 

2.  Name  the  terms  inx  —  4i/  —  Sz-\-l. 

3.  Name  the  numerical  coefficient  in  each  of  the 
monomials  2a%,,  ah^  Sabx,  mn\  Zxyz^  4(a;-|-y). 

4.  Of  the  following  monomials  name  those  that  are 
like: 

2xy,  Sx^y,  Zxy,  2xy\  \^y,  bxy\  xy,  xy\  x^y,  bxy^. 

5.  State  with  respect  to  what  letter  4  rw,  mw,  and  cm  are 
like.     What  is  the  coefficient  of  each  of  these  monomials  ? 

6.  With  respect  to  what  factor  are  axy^  bxy,  and  cxy 
like,  and  what  is  the  coefficient  of  each  monomial  ? 

7.  With  respect  to  what  factor  are  2(w  +  w),  c{m  -f-  w), 
and  d(m  +  n)  like,  and  what  is  the  coefficient  of  each 
monomial  ? 


INTRODUCTION  15 

8.  If  a  stands  for  one  number  and  h  for  another,  a-\-h 
represents  their  sum ;    a  —  b,  their  difference ;    a6,  their 

product ;  and  ^,  the  quotient  of  a  divided  by  b.     Which 

of  these  expressions  are  binomials,  and  which  monomials  ? 
Evaluate  each  when  a  =  2  and  5=1. 

Evaluate  the  following  trinomials  when  a:  =  4,  ^  =  2, 
and  2  =  1. 

9.    X  -{- 1/  -{- z.  10.    X  —  y  -\-  z.  IX.    x  —  y  —  z. 

12.    X  —  2 y  -\-  z,        13.    X  —  y  —  2z.  14.    x  +  y  —'^z, 

23.  The  equation.     Such  statements  as, 

a  =  b^  2  a;  =  3,  and  x-\-  y  =  4: 

are  called  equations.  The  equation  a  =  b  means  that  the 
number  represented  by  the  letter  a  is  equal  to  the  number 
represented  by  the  letter  b.  In  the  equation  a=b^  a  is 
called  the  first  member  of  the  equation  and  b  is  called 
the  second  member. 

24.  Application  of  the  equation. 

Problem.  $  500  is  to  be  divided  between  two  persons,  A  and  B,  in 
such  a  way  that  B  shall  receive  $  100  more  than  A.  How  many  dol- 
lars should  each  receive  ? 

Solution.         Let  x  =  the  number  of  dollars  in  A's  share. 

Then,  x  -)-  100  =  the  number  of  dollars  in  B's  share. 

Hence,  x  -\-  x  -\-  100  =  the  number  of  dollars  both  are  to  receive. 

meaning  that  two  times  the  number 
of  dollars  that  A  receives  plus  100  is 
equal  to  500,  the  number  of  dollars  to 
be  divided. 
-100, 

.'.  X  =  200,  the  number  of  dollars  that  A  is  to  receive. 
x  +  100  =  300,  the  number  of  dollars  that  B  is  to  receive. 
Note.     The  symbol .-.  is  read  therefore. 


That  is. 

2a:  +  100=.500< 

Hence, 

2x  =  500 

or, 

2a;  =  400; 

16  ELEMENTARY  ALGEBRA 

In  the  foregoing  solution,  the  equation  2x  =  500  —  100 
was  obtained  from  the  preceding  equation  *2x-\-  100  =  500, 
by  means  of  the  principle  that  if  equal  numbers  he  subtracted 
from  equal  numbers  the  resulting  numbers  are  equal.  Also, 
the  equation  x  =  200  was  obtained  from  the  equation 
2x  =  400  by  means  of  the  principle  that  if  equal  numbers 
be  divided  by  equal  numbers  (the  number  zero  excepted}  the 
resulting  numbers  are  equal. 

25.  Assumptions.  The  simplification  of  all  equations 
depends  on  such  assumptions,  or  principles,  as  those 
stated  in  section  24.     These  assumptions  are : 

1.  ^  equal  numbers  be  added  to  equal  numbers^  the  sums 
will  be  equal. 

2.  If  equal  numbers  be  subtracted  from  equal  numbers^ 
the  differences  will  be  equal. 

3.  If  equal  numbers  be  multiplied  by  equal  numbers^  the 
products  will  be  equal. 

4.  If  equal  numbers  be  divided  by  equal  numbers  (zero 
excepted)  .>  the  quotients  will  be  equal. 

5.  Numbers  which  are  equal  to  the  same  number  are  equal 
to  each  other. 

26.  The  generalizing  spirit  of  algebra  may  be  illus- 
trated by  further  consideration  of  problems  similar  to 
that  of  section  24. 

ILLUSTRATION 

Instead  of  dividing  $  500  between  two  persons,  let  the  number  of 
dollars  divided  be  represented  by  the  letter  w,  which  may  represent 
any  number  of  dollars,  and  let  the  second  person  receive  a  dollars 
more  than  the  first. 

Solution.     Let  x  =  the  number  of  dollars  in  A's  share. 

Then,  x  -\-  a  =  the  number  of  dollars  in  B's  share. 

Hence,  x  -^  x  -\-  a  —  the  number  of  dollars  they  both  receive. 


INTRODUCTION  17 

That  is,  2x  -{■  a  =  n, 

or,  2x  =  n-a,  [§25,2] 

.•.x  =  ^.  [§25,4] 

Hence,  n  —  a  _  ^j^^  number  of  dollars  in  A's  share, 

and  n  —  a  _j_  ^  _  ^j^g  number  of  dollars  in  B's  share. 

27.   The  resulting  value  of  x  in  the  solution,  section  26, 
namely,  x  =     ""    ,  is  an  illustration  of  what  in  algebra  is 

termed  a  formula.     This  formula  gives  the  solution  of  a 

great  number  of  particular  problems  which  are  all  of  the 

same  kind  and  which  differ  only  in  the  numerical  values 

assigned  to  the  letters. 

Thus,  when  n  =  500  and  a  =  100,  [§  24] 

500  -  100 


2 

and  x  +  100  =  300 ; 

which  were  the  results  obtained  in  §  24. 


=  200, 


28.   As  a  general  definition  we  have  the  following  : 

A  formula  is  a  rule  of  calculation  expressed  in  algebraic 

symbols. 

Remark.     Problem  3,  Exercise  2,  page  7,  contains  an  important 

geometrical  formula,  that  for  the  area  of  a  rectangle. 

ILLUSTRATIVE  EXAMPLES 

1.  When  a:  +  3  =  7,  what  is  the  value  of  x ;  that  is,  for 
what  number  does  x  stand  ? 

Solution.  ar  +  3  =  7. 

Subtracting  3  from  each  member,      a?  =  7  —  3.  [§  25,  2] 

.*.  X  =  4:. 

2.  When  3  a:  =  12,  what  is  the  value  of  a;? 
Solution.  8  ar  =  12. 

Dividing  both  members  by  3,  x  =  4.  [§  25, 4] 


18 


ELEMENTARY  ALGEBRA 


3.    When  Jo;  =  2,  what  is  the  value  oi  x? 

Solution.  '        J  a:  =  2. 

Multiplying  both  members  by  4,        x  =  8. 


[§  25,  3] 


4.  Solve   3  a;  +  5  =  11   and  check  the  resulting  value 
of  X. 

Solution.  3  a;  +  5  =  11. 

3a;  =  11 -5. 

3a;  =  6. 
x  =  2. 
Check.  3  X  2  +  5  =  6  +  5,  or  11. 

5.  Solve  3a;  +  2a;  — a;4-7  =  2a;  +  15  and  check. 

Solution.  3a;  +  2a;-a;  +  7  =  2a;  +  15. 

4a;  +  7  =  2x  +  15. 

[See  Example  9,  Exercise  2] 
4a;-2a;  +  7  =  2a;-2a;+15.       [§25,2] 
2a;  +  7  =  15. 
2a;  =  8. 
a;  =  4. 
Check.  3x4 +  2x4-4  +  7  =  2x4  + 15. 

12  +  8  -  4  +  7  =  8  +  15. 
23  =  23. 


EXERCISE  7 

Solve  the  following  equations  and  check : 

1.    a;  +  3-5.        •  2.  a;  +  7  =  9. 

3.    a;  H-  2  =  3.  4.  a;  +- 1  =  5. 

5.    a;  +- 1  =  4.  6.  a;  +-  7  =  8. 

7.    3  + a;  =  4.  8.  2  + a;  =  4. 

9.    4+-a;  =  6.  10.  2a;  +  3  =  7. 

11.    3a;  +  l==10.  12.  5a:H-2  =  12. 

13.    3a;4-3  =  9.  14.  5a; +-1  =  11. 

15.    2a;-h4  =  6.  l^.  3a;+-4  =  5, 


INTRODUCTION  19 

17.  42^  +  8  =  12.  18.  72:4-6  =  20. 

19.  32:+ 22:  =  10.  20.  32;-f-2;  =  12. 

21.  52:+ 22: -2^=12.  22.  62:  =22: +  3. 

23.  72:- 2:  + 1  =  22:  + 9.  24.  2x-\-Sx  —  4:X=B, 

25.  ^2:  =  3.  26.  ^2:  =  3. 

27.  ^2:=1.  28.  52:  + 1=32; +  7. 

29.  52:  +  2  =  62:+l.  30.  4x-x -\- 2x  =  ^x -{-4:. 

In  problems  31-35,  denote  the  unknown  number  by  x. 
Form  the  equation  and  find  the  value  of  2:. 

31.  Twice  a  certain  number  is  20.  What  is  the 
number  ? 

32.  The  sum  of  10  and  twice  a  certain  number  is  50. 
What  is  the  number  ? 

33.  If  three  times  a  certain  number  is  added  to  9,  the 
sum  is  15.     What  is  the  number  ? 

34.  The  sum  of  two  consecutive  integers  is  11.  Find 
the  integers. 

Suggestion.     Consecutive  integers  are  those  which  differ  by  1. 

35.  If  a  pound  of  butter  and  one  of  lard  together  cost 
60  cents,  what  was  the  price  of  each  if  the  butter  cost 
three  times  as  much  as  tlie  lard  ?  y 

36.  The  perimeter  of  the  rectangle 
in  the  accompanying  diagram  is  given 
by  the  formula  P  =  2  2:  +  2  y.  Find 
X  when  P  =  100  and  «/  =  30.  Find  t/ 
when  P  =  50  and  2:  =  10. 


When  A  stands  for  the  area  of  a  rectangle,  h  the  num- 
ber of  units  in  its  base,  and  a  the  number  of  units  in  its 
altitude,  A  =  axb. 


20  ELEMENTARY  ALGEBRA 

Using  this  formula,  find  the  value  of  the  missing  letter, 
given : 

37.    6  =  6,  a  =  4.  38.    6  =  10,  a  =2^. 

39.    ^  =  20,  a  =  4.  40.    A  =  60,  ,b  =  10. 

When  A  stands  for  the  area  of  a  triangle,  b  the  number 
of  units  in  its -base,  and  a  the  number  of  units  in  its 
altitude,  ^  _  a  x  6 

From  this  formula  find  the  value  of  the  missing  letter, 
given: 

41.    a  =  6,b  =  8.  42.    a=9,b  =  4. 

43.    ^  =  12,  a  =  6.  44.    A  =  20,b  =  S. 

When  r  stands  for  the  number  of  units  in  the  radius  of 
a  circle,  c  the  number  in  the  circumference,  and  A  the 
area,  then,        (j)  ^  ^  2  irr;      (2)  A  =  irr^. 

Using  the  proper  formula, 

45.  Find  (?,  given  r  =  3. 
Thus,  c  =  2  TT  X  3,  or  6  TT. 

46.  Find  A,  given  r  =  5, 
Thus,  ^  =  -TT  X  52,  or  25  tt. 

47.  Find  r,  given  c  =  8  tt. 

Thus,  2  7rr=8ir;  2r  =  8;  r  =  4. 

Remark.  Observe  that  the  equation  2  r  =  8  was  derived  from  the 
equation  2  irr  =  8  tt  by  dividing  both  members  of  the  latter  by  x. 

48.  Find  r,  given  A  =  9  tt. 

Thus,  irr2  =  9  TT ;  r2  =  9 ;  r  =  3. 

49.  Find  c,  given  r  =  6.  50.  Find  A,  given  r  =  2. 
51.  Find  A,  given  r  =  10.        52.  Find  r,  given  c  =  16  tt. 
53.  Find  r,  given  ^  =  647r.       54.  Find  r,  given  -4  =  |  tt. 


Introduction  21 

55.  Using  the  letters  c  and  n,  write  a  formula  for  the 
cost  of  any  number  of  dozen  oranges  when  one  dozen 
costs  25  cents ;  when  one  dozen  costs  a  cents. 

56.  Write  as  a  formula  the  rule  for  finding  the  simple 
interest  for  a  given  number  of  years  i  on  a  given  sum  of 
money  s  at  a  given  rate  of  interest  per  cent  per  annum  r. 

57.  I  am  twenty-five  years  younger  than  my  father, 
whose  age  is  a  years.  Write  in  a  formula  the  rule  for 
finding  my  age  when  his  age  is  known. 

58.  Write  a  formula  for  the  weight  of  a  bottle  of  milk, 
given  the  weight  of  the  bottle,  the  weight  of  a  cubic  inch 
of  milk,  and  the  quantity  in  the  bottle.  (Use  the  letters 
Tf,  b,  if,  and  v.) 

59.  Write  a  formula  for  the  number  of  cents  in  a 
dollars  +  h  dimes  -h  e  nickels. 

60.  Write  a  formula  for  the  number  of  inches  in  a  yards 
b  feet  c  inches. 

61.  Write  an  expression  for  the  rate  of  a  train  which 
runs  m  miles  in  5  hours ;  which  runs  m  miles  in  h  hours. 

62.  If  n  articles  cost  d  dollars,  find  an  expression  for 
the  number  of  articles  which  can  be  bought  for  x  dollars. 

63.  Construct  a  formula  for  the  number  iV  which,  when 
divided  by  d,  gives  the  quotient  q  and  the  remainder  r. 

64.  The  sum  of  three  consecutive  integers  is  18.  Find 
the  integers. 

65.  Write  a  formula  for  the  volume  Fof  a  beam,  I  feet 
long,  w  feet  wide,  and  d  feet  thick.  Apply  the  formula 
to  find  the  thickness  of  a  beam  which  contains  10  cubic 
feet  of  timber  and  is  2  feet  wide  and  5  feet  long. 

66.  What  is  the  area  of  a  triangle  whose  base  is  5  feet 
and  whose  altitude  is  10  feet  ? 


22  ELEMENTARY  ALGEBRA 

67.  A  room  is  a  feet  long  and  h  feet  wide.  Construct 
a  formula  for  the  cost  of  carpeting  the  room  with  lino- 
leum which  costs  $2  per  square  yard. 

68.  The  rule  for  making  tea  is  :  "  One  teaspoonful  of 
tea  for  each  person  and  one  for  the  pot."  Express  this 
rule  by  a  formula  letting  t  denote  the  number  of  tea- 
spoonfuls  of  tea  and  p  the  number  of  persons. 

69.  A  formula  for  the  time  in  hours  required  to  cook  a 
joint  of  beef  of  given  weight  in  pounds  is  t  =  \w  +  \. 
From  this  formula  state  the  rule. 

Positive  and  Negative  Numbers 

29.  Algebra  makes  use  of  all  the  numbers  of  elementary 
arithmetic  and  in  addition  to  these  it  introduces  certain 
other  numbers  called  positive  and  negative  numbers. 

ILLUSTRATION 

Let  a  point  on  a  horizontal  line  be  selected  and  marked  0 ;  let  the 
point  one  unit  to  the  right  of  0  be  marked  +  t,  the  point  two  units 
to  the  right  of  0  be  marked  +  2,  and  so  on.  Let  the  point  one  unit 
to  the  left  be  marked  —  1,  the  point  two  units  to  the  left  of  0  be 
marked  —  2,  and  so  on ;  thus : 

I I  I I I I I I I 


_4     _3     _2     -1         0     +1     +2     +3     +4 
The  points   marked    +1,    +  2,  +  3,   etc.,   represent  the  positive 
numbers,  and  those  marked  —  1,-2,  —  3,  etc.,  represent  the  negative 
numbers. 

30.  Two  positive  or  two  negative  numbers  are  said  to 
have  like  signs,  A  positive  number  and  a  negative 
number  are  said  to  have  unlike  signg.  Positive  and 
negative  numbers  are  called  algebraic  numbers, 

31.  Whenever  quantities  exist  which  are  exact  opposites, 
as  illustrated  in  section  29,  these  quantities  may  be  repre- 
sented by  positive  and  negative  numbers. 


INTRODUCTION  23 

Thus,  an  ordinary  thermometer  scale  may  be  divided  in  the 
manner  indicated  by  the  diagram  in  section  29.  When  so  divided 
the  point  marked  0  indicates  zero  degrees ;  one  degree  above  zero  is 
marked  +  1,  two  degrees  above  zero,  -f  2,  onjp  degree  below  zero, 
—  1,  two  degrees  below  zero,  —  2,  and  so  on. 

32.  The  scale  of  positive  and  negative  numbers,  section 
29,  has  various  practical  applications,  such  as  indicating 
degrees  of  latitude  north  '  and  south  from  the  earth's 
equator,  marked  0°;  degrees  of  longitude  west  and  east 
from  some  chosen  meridian,  as  that  of  Greenwich,  which 
is  marked  0°  ;  intervals  of  time  after  and  before  a  certain 
event ;  gains  and  losses  in  business  transactions. 

Note.  Negative  numbers  are  introduced  into  algebra  by  a  simple 
convention.  In  a  -  6  it  is  convenient  to  call  the  expression  —  6  a 
number  and  to  say  that  a  —  b  is  obtained  by  adding  —  6  to  «.  Thus, 
a  —  b  is  written  a  -\-{—  b).  This  is  a  new  use  of  the  word  number. 
A  negative  number  is,  therefore,  simply  a  number  which  is  to  be  sub- 
tracted. In  the  same  way  a  positive  number  is  a  number  which  is 
to  be  added.  The  numbers  of  arithmetic  are  neither  positive  nor 
negative. 

EXERCISE  8 

1.  Using  the  signs  +  and  — ,  write  : 

5  positive  units ;  6  negative  units ;  a  positive  units ; 
h  negative  units ;  2  a  positive  units ;  Sx  negative  units. 

2.  State  how  many  and  what  kind  of  units  there  are  in 
each  of  the  following  : 

-1-3;    -1;    +c;   -b;    ■j-2x;    -Sb. 

3.  If  10°  north  latitude  is  represented  by  +  10°,  what 
number  will  represent  25°  north  latitude  ?  10°  south 
latitude  ?     30°  south  latitude  ?  ^ 

4.  If  north  latitude  is  marked  -f-  and  south  latitude  — , 
write,  using  the  signs  +  and  —  instead  of  N.  and  S. : 

6°  N.;  6°  S.;  9°  30'  N.;  7°  30'  12"  S. 


24  ELEMENTARY  ALGEBRA 

5.    If  west  longitude  is  marked  -|-  and  east  longitude 
— ,  write,  using  the  signs  +  and  —  instead  of  W.  and  E. : 
20°  E.;  5°  W.;  4°  15'  W. ;  7°  10'  20''  E. 

6.'  If  the  year  1916  a.d.  is  represented  by  +1916,  what 
will  represent  the  year  2000  A.D.?  The  year  399  B.C.? 
The  year  646  B.C.? 

7.  If  gains  are  marked  +  'and  losses  — ,  write,  using 
the  signs  +  and  — : 

15  gain;  $6  loss;  18  loss:  $4  gain. 

8.  If  temperature  above  zero  is  marked  -|-  and  tempera- 
ture below  zero  — ,  state  what  temperature  is  indicated  by 
each  of  the  following  : 

^5°;    ^lo.    +60°;    -7°;    +80°. 

Addition 

33.  A  gain  of  f  10  together  with  a  gain  of  $6  makes  a 
total  gain  of  f  16  ;  also  a  loss  of  -f  10  together  with  a  loss 
of  $6  makes  a  total  loss  of  $16.  Hence,  regarding  gain 
as  positive  and,  therefore,  loss  as  negative^  we  may  infer  that : 

1.  (+10)  +  (+6)=  +  16. 

2.  (_10)4.(-6)  =  -16. 

Remark.  If  it  is  necessary  to  distinguish  a  sign  of  an  algebraic 
number  from  a  sign  of  operation,  the  algebraic  number  is  put  into 
parentheses ;  otherwise,  in  writing  such  numbers  the  positive  sign  is 
usually  omitted. 

34.  A  gain  of  $10  combined  with  a  loss  of  $6  is  equiva- 
lent to  a  net  gain  of  f  4 ;  also,  a  loss  of  -f  10  combined  with 
a  gain  of  $6  is  equivalent  to  a  net  loss  of  |4.  Hence,  we 
may  infer  that : 

1.  (+10)  +  (-6)  =  -f-4. 

2.  (_10)-h(+6)  =  -4. 


INTRODUCTION  25 

35.  The  absolute  value  of  an  algebraic  number  is  its 
value  without  regard  to  sign. 

Thus,  the  absolute  value  of  +  2  is  2  and  the  absolute  value  of  —  3  is  3. 
Remark.     The  expressions  arithmetical  value  and  numerical  value  are 
sometimes  used  instead  of  absolute  value. 

36.  Positive  and  negative  numbers  are  also  called 
opposite  numbers.     See  section  31. 

37.  The  result  obtained  by  combining  (adding)  two  or 
more  algebraic  numbers  is  called  the  sum  of  the  numbers. 

From  sections  33  and  34  we  infer  that: 

1.  The  sum  of  two  numbers  with  like  signs  is  the  sum  of 
their  absolute  values  with  their  common  sign  prefixed  to  the 
result. 

2.  The  sum  of  two  numbers  with  unlike  signs  is  the  dif- 
ference of  their  absolute  values  with  the  sign  of  the  number 
which  has  the  greater  absolute  value  prefixed  to  the  result. 

Note  1.  To  add  three  or  more  algebraic  numbers  with  like  signs, 
add  the  second  to  the  first,  to  the  result  add  the  third,  and  so  on. 

Thus,    +3+(+2)+(+4)  +  (+l)  =  +5  +  (+4)  +  (+l) 

=  (+9)  +  (+l)  =  (+10). 

Note  2.  To  add  three  or  more  algebraic  numbers  with  unlike 
signs,  add  the  positive  numbers  and  the  negative  numbers  separately, 
and  then  add  the  results. 

Thus,  +4+(+o)  +  (+6)  +  (-2)-f-(-7)=-|-15+(-9)=  +  6. 

Remark.  When  two  numbers  have  the  same  absolute  value  and 
unlike  signs,  their  sum  is  zero. 

Thus,  (+2) +  (-2)  =0. 


EXERCISE  9 

Name  at  sight  the  sum : 

1.     +3                 2.-2                 3. 

+  5 

4. 

-4 

+  2                    -1 

+  7 

-9 

26  ELEMENTARY  ALGEBRA 

5. 


+  2 

6.    +3 

7.-8 

8.-6 

-1 

-5 

+  5 

+  9 

9.-4  10.    -10  11.    +7  12.    +8 

+  4  -    1  -6  -8 

Add,  as  indicated,  at  sight : 

13.    +5 +  (+2).  14.    -7 +  (-3). 

15.    -8 +  (-6).  16.    -8 +  (-3). 

17.    +9  +  (_9).  18.    +10 +  (-12). 

19.    +2+(+l)  +  (+4).       20.    _2-f-(-3)  +  (-4). 

21.    -4  +  (+l)  +  (+2).       22.    +6  +  (-2)  +  (-l). 

23.  _l  +  (_2)  +  (+l)  +  (+3). 

24.  +2  +  (-3)  +  (4-4)  +  (-l). 

25.  _l+(+l)  +  (+5)  +  (-3). 

26.  +3  +  (-3)  +  (-4)  +  (+4). 

27.  _2+(-3)  +  (+4)  +  (-3). 

28.  6  +  (-3)  +  (+2)  +  (-7). 

29.  5+(-|.3)  +  (-4)  +  (-8). 

30.  _7+(-2)  +  (+8)  +  (+2). 

Subtraction 

38.  In  algebra,  as  in  arithmetic,  subtraction  is  the 
inverse  operation  of  addition ;  that  is,  subtraction  is  the 
undoing  of  an  addition. 

Thus,  5  +  3-3  =  5,  which  shows  that  the  addition  of  3  has  been 
undone  by  the  subtraction  of  3. 

Also,  a  +  3  —  3  =  a,  in  which  a  is  any  algebraic  number. 

In  general,  the  result  of  subtracting  a  number  from  an 
equal  number,  whether  it  be  positive  or  negative,  is  zero. 
Thus,  a  -  a  =  0,  since  0  +  a  -  a  =  0. 


INTRODUCTION  27 

Since  a  —  a=  0  and  a  -h (  —  «)  =  0  [section  37,  remark], 
the  following  principle  may  be  inferred : 

The  result  of  subtracting  a  number  is  the  same  as  the 
result  of  adding  the  opposite  number. 

ILLUSTRATIVE  EXAMPLES 

1.  Subtract  +  3  from  +  5. 

Solution.     (+  5)-(+  3)  =  (+  5)  +  (-  3)  =  +  2. 

2.  Subtract  —  3  from  +  5. 

Solution.     (+5)-(-3)  =  +5+(+3)=  +  8. 

3.  Subtract  +  7  from  +  2. 

Solution.     (+2)-(-|-7)  =  + 2 +(-7)  =  - 5. 

4.  Subtract  +  3  from  —  5. 

Solution.     (-5)-(+3)  =  - 5+(-3)=-8. 

5.  Subtract  —  2  from  —  5. 

Solution.     (- 5)-(-2)  =  - 5+(+2)  =  -3. 

Remark.  In  algebra  the  terms  minuend,  subtrahend,  and  difference 
have  the  same  meaning  as  in  arithmetic. 

Thus,  in  (+  6)  —  (+  2)  =  +  4,  the  minuend  is  +  6,  the  subtrahend 
is  +  2,  and  the  difference  is  +  4. 

39.  From  the  solutions  of  the  illustrative  examples  of 
section  38  the  following  rule  may  be  derived: 

Rule.  The  difference  of  two  algebraic  numbers  is  found  by 
adding  to  the  minuend  the  subtrahend  with  its  sign  changed. 

EXERCISE   10 

Name  at  sight  the  difference : 

1.    +  5  2.-5  3.    +  3  4.-4 

+  2  -3  H-7  -9 


28  ELEMENTARY  ALGEBRA 


5.     +9                 6.-7                 7.     - 

-9 

8.    4-1 

-2                    4-3 

-1 

+  8 

9.-6              10.-6            11. 

0 

12.          0 

-6                       +6 

4-8^ 

-5 

Subtract,  as  indicated,  at  sight : 

13.    +6 -(4- 2).     14.    -4 -(-3). 

15. 

4-7-(-l). 

16.    -9 -(4- 5).     17.    -1-(4.1). 

18. 

-l-(-7). 

19.    11 -(-4).        20.    7 -(-10). 

21. 

0-(4-5). 

22.    O-(-l).          23.    6 -(4- 5). 

24. 

5 -(-5). 

40.   In  section  33  it  is  shown  that 

10  4-(4-6)=16  =  10  4-6, 

(1) 

also  that              10  4- (-  6)  =  4  =  10  - 

-6. 

(2) 

Again,  from  section  39  we  have 

10-(4-6)=.10+(- 

-6)  = 

10  -  6,         (3) 

also  that             10 -(- 6)=  10  4-(4- 6)  = 

10  4-6.         (4) 

Examples  (1),  (2),  (3),  and  (4)  illustrate  the  following : 

Rule  of  Signs 

4-  4-  or  —   —  mat/  be  replaced  by  4-. 
4-  —  or  —   4-  may  be  replaced  by  — . 

From  examples  (1),  (2),  (3),  and  (4),  it  is  evident  that 
in  addition  a  number  is  written  down  with  the  sign  before 
it  retained,  while  in  subtraction  the  number  is  written 
with  its  sign  changed. 

Thus,  (+  2)  +  (+  3)  +  (  -  4)  is  written  2  +  3-4  =  1; 

but,  (  +  2)  -  (-  6)  is  written  2  +  6  =  8. 

Also,  -  4  +(  -  3)  is  written  -  4  -  3  =  -  7 ; 

but,  -  4  -  (-  3)  is  written  -  4  +  3  =  -  1. 


INTRODUCTION  29 

Remark.  An  expression  like  4  -  6  =  —  2  does  not  occur  in 
arithmetic.  In  algebra,  this  expression  means  simply  [see  diagram, 
§  29]  that  if  we  start  at  the  point  marked  +  4  and  count  6  units  to 
the  left  we  shall  end  with  the  point  marked  —  2. 

ILLUSTRATIVE  EXAMPLES 

1.  Simplify     (4.2)  +  (-3)  +  (-f4). 

Solution.     (+2)  +  (-3)+(+4)=2-3  +  4=-l  +  4  =  3. 

2.  Simplify     (-}.2)4-(-3)  +  (-4). 

Solution.       (+2)  +  (-3)  +  (-4)=2-3-4  =  -l-4=--5. 

EXERCISE  11 

Perform  the  indicated  operations  in  the  following  ex- 
amples : 

1.   (+2)-(+l).  2.  3 -(+2). 

3.   -5  +  (-2).  4.   (-2)  +  (H-5). 

5.  2 -(-3).  6.   -7 +(-2). 

7.    _2-(+l).  8.   -3-h(-5). 

9.    -2  +  (-l).  10.  3  +  (-2). 

11.  3 -(+5).  12.  3  +  5-(+2). 

13.  (_2)  +  (-3)-(-4>         14.  (-3)+(  +  2)-(-l). 
15.   -4'+(-4).  16.    -2  + 2 +  (-2). 

17.   -5  +  4+(-l).  18.   -l-2-(+3). 

19.  l  +  2  +  (-3).  20.  1-1  + 2 +  (-2). 

21.   -3 +  4 -(-5).  22.    -l_2  +  (-3). 

23.  _(_2)4-C-3)-(-l). 

24.  l4.(+9)-(+12)H-(-3)  +  (+17)-(+12). 

25.  2  +  (-9)-(-8)  +  (-l)-(+10)  +  (+20). 

26.  8-(-7)  +  (-14)-(-5)-(4-12)4-(+7). 

27.  -6-(-10)  +  (-f  20)-(4-15)  +  (-25). 


30  ELEMENTARY  ALGEBRA 

Graphic  Representation  of  Addition  and  Subtraction  of 
Algebraic  Numbers 

41.  It  is  not  possible  in  arithmetic  to  subtract  a  number 
from  a  less  number.  The  introduction  of  negative 
numbers  makes  it  possible  in  algebra  to  subtract  in  all 
cases.  The  diagram  of  section  29  is  here  reproduced  and 
used  in  illustrating  the  addition  and  subtraction  of  alge- 
braic numbers. 

I         I         I I         I I         I I         I I         I 

_5     _4     _3     _2     -1         0     +1     +2     +3     +4     +5 

1.  To  add  a  positive  number. 

To  add  +  2  to  +3  begin  at  the  point  marked  +  3  and  count  two 
spaces  to  the  right,  ending  at  the  point  marked  +  5.  In  like  manner, 
to  add  +  2  to  any  positive  or  negative  number  begin  with  that 
number  and  count  two  spaces  to  the  right.     In  general, 

In  adding  a  positive  number^  the  counting  is  to  the  right, 

2.  To  subtract  a  positive  number. 

Since  subtraction  is  the  undoing  of  an  addition,  to  subtract  +  2 
from  +  5,  begin  at  the  point  marked  +  5  and  count  two  spaces  to 
the  left,  ending  at  the  point  marked  +  3.     In  general. 

In  subtracting  a  positive  number^  the  counting  is  to  the  left, 

3.  To  add  a  negative  number. 

Since  the  sum  of  —  2  and  —  3  is  equal  to  —  5  [§  33],  to  add  —  2  to 
—  3,  begin  at  the  point  marked  -  3  and  count  two  spaces  to  the  left, 
ending  at  the  point  marked  —  5.     In  general. 

In  adding  a  negative  number^  the  counting  is  to  the  left. 

4.  To  subtract  a  negative  number. 

Since  subtraction  is  the  undoing  of  an  addition,  to  subtract  —  2 
from  —  5,  begin  at  the  point  marked  —  5  and  count  two  spaces  to  the 
right,  ending  at  the  point  marked  —  3.     In  general, 

In  subtracting  a  negative  number,  the  counting  is  to  the 
right. 


INTRODUCTION  31 

Remark.  Notice  that  in  the  two  operations  of  adding  a  positive 
and  subtracting  a  negative  number,  the  counting  is  done  to  the 
right ;  while  in  the  two  operations  of  adding  a  negative  number  and 
subtracting  a  positive  number,  the  counting  is  done  to  the  left.  Hence, 
The  subtraction  of  a  number  and  the  addition  of  its  opposite  number 
lead  to  the  same  result. 

EXERCISE   12 

Using  the  diagram  in  section  41,  verify  the  following  by 
counting : 

1.    (_2)-h(+2)=0.  2.  (-5)  +  (+9)=  +  4. 

3.    (_2)-(+2)=-4.  4.  (-5)-(-l)=-4. 

5.    0  +  (+3)=-f3.  6.  0-(+3)=-3. 

7.    0+(-3)=-3.  8.  0-(-3)=  +  3. 

9.    0  +  0  =  0.  10.  0-0  =  0. 

11.    (-l)  +  (_2)=-3.  12.  (_4)-C+l)  =  -5. 

13.    (-f-3)-H(-4)=-l.  14.  (+5)-(+8)=-3. 


Multiplication 

42.  In  arithmetic,  3x2  means  2  +  2  +  2.     In  algebra, 

1.  (+3)x(+2)  means +2+(+ 2)  +  (+2);  that  is, 
multiplication  by  a  positive  integer  means  that  another 
number  is  to  be  repeated  positively. 

2.  (-3)x(  +  2)  means  -(  +  2)-(  +  2)-(  +  2);  that 
is,  multiplication  by  a  negative  integer  means  that  another 
number  is  to  be  repeated  negatively. 

Note.  (+  3)  X  (+  2)  and  (-  3)  x  (+  2)  are  usually  written 
(+  3)(+  2)  and  (-  3)  (  +  2),  respectively. 

43.  The  number  repeated  is  called  the  multiplicand; 
the  number  which  shows  how  many  times  the  multiplicand 
is  repeated  is  called  the  multiplier;  the  result  in  multipli- 
cation is  called  the  product. 


32  ELEMENTARY  ALGEBRA 

44.  The  possible  combinations  of  signs  in  algebraic 
multiplication  are  given  in  the  following : 

1.  (+3)(+2)  =  +  (-f  2)  +  (+2)  +  (+2)=2  +  2  +  2=4-6. 

2.  (+3)(-2)  =  +  (-2)  +  (-2)  +  (-2)=-2-2-2=-6. 

3.  (-  3)(+  2)=  -(+  2)-(+  2)-  (+  2)  =  -  2  -  2  -  2  =  -6. 

4.  (-3)(-2)  =  -(-2)-(-2)-(-2)  =  +2  +  2  +  2=+6. 

45.  From  1,  2,  3,  and  4  of  section  44  we  may  infer  the 

Rule  of  Signs  in  Multiplication 

The  product  of  two  numbers  with  like  signs  is  positive; 
the  product  of  two  numbers  with  unlike  signs  is  negative. 

EXERCISE  13 

Name,  at  sight,  the  product : 

1.  (  +  3)(  +  2).         2.  (  +  4)(  +  3).  3.  (+3)(-2). 

4.  (-4)(+3).         5.  (-3)(-2).  6.  (-1X-2). 

7.  (  +  1)(-1).         8.  (-1)(-1).  9.  (  +  2)(-3). 

10.  (-3)(-3).       11.  (-5)(  +  2).  12.  (+7)(-3). 

13.  (+2)(+3)(-4). 

Suggestion.  Multiply  +3  by  +  2,  then  find  the  product  of  this 
result  and  —  4. 

14.  (+2)(+3)(-2).  15.  (_2)(-3)(-4). 
16.  (_2)(+3)(-5).  17.  (+5)(-7)(-2). 
18.   (_1)(+1X-1).  19.    (+l)(_l)(+l). 

20.  In  finding  the  product  of  three  factors,  what  is  the 
sign  of  the  product  when  only  one  of  the  factors  is  nega- 
tive? When  all  of  the  factors  are  negative?  When  two 
of  the  factors  are  negative  ? 


Franciscus  Vieta  (Francois  Viete)  (1540-1603)  was  a  French 
lawyer  who  gave  up  most  of  his  leisure  to  mathematics.  He  was 
the  author  of  the  earliest  work  on  symbolic  algebra.  He  introduced 
in  this  work  the  use  of  letters  for  known  and  unknown  numbers. 
His  solution  of  the  cubic  equation  continues  in  use  at  the  present 
time. 


INTRODUCTION  33 

Division 

46.  In  arithmetic,  12  (the  dividend)-!-  3  (the  divisor) 
=  4  (the  quotient),  because  8  x  4  =  12. 

That  is,  2)/i;/5o;.  X  Quotient  =  Dividend. 

This  relation  connecting  divisor,  quotient,  and  dividend 
may  be  employed  in  establishing  the  rule  of  signs  in 
division,  thus  : 

1.  (+12)-f-(+4)  =  +3,  since(+4)(+3)=  +  12. 

2.  (+12)-(-4)=-3,  since  (- 4)(- 3)=  +  12. 

3.  (-12)-(+4)=:-3,  since  (+4) (-3)  =  -12. 

4.  (-12)-(-4)  =  -f  3,  since  (-4)(+3)  =  -12. 

From  1,  2,  3,  and  4,  we  may  infer  the 

Rule  of  Signs  in  Division 

77ie  quotient  of  two  numbers  with  like  signs  is  positive; 
the  quotient  of  two  numbers  with  unlike  signs  is  negative. 


BXBRCISI 

:  14 

Name,  at  sight,  the  quotient : 

1.    (+6)^(+2). 

2. 

(+9K(+3). 

3.    (+6)^(-2). 

4. 

(-6)  +  (+2). 

5.    (-8)^(+l). 

6. 

(_3)^(-l). 

7.    (-24)^(-12). 

8. 

(-27)^  (-3). 

9.    (-14)^(-2). 

10. 

(+27)^(+9). 

11.    (+30)h-(-8). 

12. 

(+56)^(-8). 

13.  (+12)--(-t-3)^(+2). 

Suggestion.     Divide  (+  12)  by  (+  3),  then  divide  the  resulting 
quotient  by  (+  2). 

14.  (+24)  +  (-2)H-(-3> 

15.  (_18)^(-3)-v(-3). 


CHAPTER   II 

FUNDAMENTAL   PROCESSES 

Addition 

47.  Addition  of  monomials  which  have  no  common  factor. 

1.  The  sum  of  3 a,  2 5,  and  5c  is  -\-  Sa -\-  2b  -\-  5c. 

2.  The  sum  of  3«,26,and  -5cis  +Sa -{-2b +  (- 5  c), 
which  is  equal  to  H-3a4-25  —  5e. 

In  general, 

The  sum  of  two  or  more  monomials  which  have  no  common 
factor  is  expressed  by  writing  the  monomials  in  turn,  each 
preceded  by  its  own  sign. 

Remark.  For  convenience,  a  plus  sign  before  the  first  term  is 
usually  omitted. 

Thus,  +3a  +  2&  +  5cis  written  3  a  +  2  ft  +  5  c. 

48.  Commutative  law  for  addition.  In  arithmetic,  it  is 
not  necessary  to  call  attention  to  the  obvious  fact  that, 
for  instance,  2  -}-  3  =  3  +  2.  In  algebra,  it  is  assumed 
that  the  sum  does  not  depend  on  the  order  in  which  the 
terms  are  taken;  this  assumption  is  usually  referred  to 
as  the  commutative  law  for  addition.  This  law  is  ex- 
pressed by  the  formula, 

a^h=b-\-  a. 

EXERCISE   16 

Name  the  sum  in  each  of  the  following : 
1.    a  and  c.  2.    x  and  1. 

3.    b  and  <?.  4.    4  and  a. 

34 


FUNDAMENTAL  PROCESSES  35 

5.    «,  5,  and  c.  6.    2  a,  —  5,  and  —  c. 

7.    4  a;,  2^,  and  —  2.  8.    —  4  m,  —  5  ii,  and  p, 

9.  3(a  +  6)and  -4(<?4-c?). 

10.  2(2:  +  ^).  and  3(m  —  n). 

11.  a,  5,  <?,  and  —  c?. 

12.  3(a— 5)  and  4(a  +  6). 

13.  B^,  r\  and  Rr. 

14.  4  r,  -  2  y,  2/,  and  -  3  i. 

49.   Addition  of  monomials  which  have  a  common  factor. 

1.  2a  +  3a  =  (2  +  3)a  =  5a.  [See  problem  9,  Exer- 
cise 2,  page  8.] 

2.  2aH-3a-4«=(2-f3-4)a  =  a. 

3.  2a  +  3a-4a-5a  =  (2  +  3-4-5)a=-4a. 
In  general, 

The  sum  of  two  or  more  monomials  which  have  a  common 
factor  is  equal  to  the  sum  of  the  coefficients  of  the  common 
factor  multiplied  by  the  common  factor . 

EXERCISE   16 

(Solve  as  many  as  possible  at  sight.) 


Add: 

1. 

2. 

3. 

4. 

4:X 

—  4m 

—     m 

Sab 

3rr     . 

'  —  5m 

—  5m 

4ab 

5. 

6. 

7. 

8. 

-2a 

3mri 

9a% 

—  4a6c? 

5a 

—  8mw 

-Sa% 
11. 

9abc 

9. 

10. 

12. 

-%a^ 

irBP' 

2(a  +  b) 

ab(m-\-ny 

^7^ 

AirB^ 

3(a  +  5) 

-3a6(m+7i)2 

36  ELEMENTARY  ALGEBRA 


13. 

14. 

15. 

16. 

3a 

-6m2 

2m7i2 

-     xyz 

4a 

-8m2 

3mn2 

-4:xyz 

5a 

-5m2 

5  mw2 
19. 

-2xyz 

17. 

18. 

20. 

-4m 

-5a 

—     mw 

-  18  xyz 

5  m 

7a 

—  3mw 

12  xyz 

3m 

-3a 
22. 

—  bmn 

xyz 

21. 

23. 

24. 

-      :r2 

-  10  pq 

-    Its 

aW 

3a:2 

pq 

9^8 

-  8  ah^c^ 

2^2 

-  ^^P^ 

-      ^« 

-  6  a62tf2 

-5a^ 

-  l^pq 

-  11  ts 

9a62c2 

25 

26. 

27. 

28. 

2(a 

+  *) 

-  (m  -f 

71) 

(^  + 

1) 

3(rr  +  ;5^)2 

-4(a 

+  h} 

-3(m  + 

n) 

-  2(r  +  1) 

-4(a:H-y)2 

5(a 

+  *) 

5(m  + 

n) 

-8(r4- 

1) 

-  8(0.  +  yY 

-  6(a 

+  ^) 

4(m  + 

n) 

6(r  +  1) 

ii(^  +  yy 

29.  2  a;,  3  a;,  and  5  x. 

30.  —  mn,  —  2  mw,  and  —  5  mn, 

31.  —  4  w,  5  w,  and  —  n. 

32.  ^2^,  _  5  52^^  and  9  62^. 

33.  -  R,  -6R,  and  8  7?. 

34.  4  /,e5    _  8  5^^  and  -  3  brA 

35.  m2n,  —  3  m2n,  —  m2w,  and  4  m2n. 

36.  2  7ri22,  _  4  ^i22^  7ri22,  and  3  wB^. 

37.  —  6  a;y2^  10  xy^^  —  xy^^  and  —  4  xy^. 

38.  —  J  iB2,  :|  a^,  —  x^^  and  |  a:2. 


FUNDAMENTAL  PROCESSES  37 

39.  (w  +  w),  —  3(m  +  7i),  and  —  ^(m  -f  n). 

40.  a(2:.  +  y)\  -  <o  a{x  +  y^,  8  a(x  +  2^)^. 

Just  as  the  sum  of  3  m  and  5  m  may  be  written  (3  +  5)  m,  so  the 
sum  of  am  and  2  m  is  written  (a  +  2)  ?w.  Similarly,  am  +  6m  =  (a  +  5)/w. 
aw  +  bm  —  m  =(a  +  6  —  l)/w,  and  c(m  +  n)  +  d  (m-^ n)  =  (c -\- d)(m-^ n). 

Add  by  combining  the  coefficients : 
41.    cy  and  by.  42.    aa;  and  —  x. 

43.    a5^  and  (?6y.  44.    mxb  and  —  cyJ. 

45.    ^  aB  Siud^  ab,  46.    ^  J?LB  and  —  J  iTJ. 

47.    am^  —  m,  and  <?w.  48.    ttB^,  tt/^,  and  7ri2r. 

49.  i  HttB^ -\- i  Rirr^ -h  i  RirRr, 

50.  a(2:  +  y)  and  ^  b(^x+  y^. 

51.    ^(?/ +  z)  and  r(y  +  z).      52.    a(a:  — y)  and  (a;  —  y). 

50.  Addition  of  monomials.     The  sura  of  2  a,  3  5,  —  3  a, 

5  c,  2  6,  —  6  a,  and  7  b  may  be  found  as  follows  : 

2a  +  36-H<-3a)+5c  +  26+(-6a)+7  6 

=  2a  +  3&-3a  +  5c  +  26-6a  +  76  [§40] 

=  (2a-3a-6a)+(3  6  +  2&  +  7&)+5c  [§48] 

=  (2  -  3  -  6)a  +(3  +  2  +  7)6  +  5c  [§  49] 

=  -7a  +  12  6  +  5c. 
In  general, 

TAg  s-WTw  of  two  or  more  monomials  is  obtained  by  finding 
the  sum  of  those  that  have  a  common  factor  and  then  adding 
these  sums. 

51.  Associative  law  for  addition.  In  arithmetic,  it  is  not 
necessary  to  call  attention  to  the  obvious  fact  that,  for 
instance,  2  +  3  +  1  +  6=  (2 +3) +  (1  +  6)  =  5  +  7  =  12. 
In  algebra,  it  is  assumed  that  the  sum  does  not  depend  on 
the  grouping  of  terms ;  this  assumption  is  usually  referred 
to  as  the  associative  law  for  addition.  This  law  is  ex- 
pressed by  the  formula, 

fl  +  (&+c)  =  (a+6)  +  c. 


38  ELEMENTARY  ALGEBRA 

EXERCISE   17 

Add: 
1.  a  and  h.  2.  a  and  —h.  3.  a  and  3  h, 

4.  2  a  and  3  5.       5.   3  a  and  —^x.    6.   a,  h,  and  c. 
7.  a:,  ^,  and— z.    8.  m,  —  ti,  andp.    9.   2  a,  3  5,  4  a,  and  55. 

10.  ^x,  4a^,  -  5a;,  and  -%x^. 

11.  —  3  a,  4  5,  5  a,  —  c,  4:c^  and  —  a. 

12.  —  a;^,  2  rrz,  3  yz,  4  xt/^  5  ?/z,  and  —  6  a:2. 

13.  3  m^,  —  mn,  2  ^2^  3  mn,  —2n\  4  w^,  and  5  mn. 

14.  -  3^2^  22:?/,  5^2^  22^,  -  4a:«^,  and  62^2, 

15.  a\  -  2  a5,  iX  -a^  -2  ah,  and  -  52. 

16.  4a25,  2ab,  -  bah\  -  6a%  2a%  -ah^,  -iah,  and 
8  a25. 

17.  2(w  +  n),  3(m  —  w),  —  5(m  +  n),  and  —(m  —  n). 

18.  4(2^  +  2^),  _4(a;-y),  -2(a:-y),  -(a:  +  y), 
-  3(a;  +  ?/),  and  -  (a;  -  3/). 

19.  2(a;-3^),  52,  -4,  -Cx-y\  -7,  -452,  and  (a:-y). 

20.  a25,  a52,  3  ab,  4  ^25,  _  6  ^25^  _  5  ^52^  and  -  ab. 

21.  a2,  a,  (a  H-  6),  —  3  a,  —  4  a2,  —  5(a  +  5),  and 
4(a  +  5).. 

22.  x^i/  +  3  a;y  —  2  a;y2  4.  5  a^t/  —  xy^  —  \xy-\-2  x^y. 

52.  Addition  of  polynomials.  Since  a  polynomial  is  the 
sum  of  two  or  more  monomials,  no  additional  rules  are 
necessary  for  finding  the  sum  of  two  or  more  polynomials. 
However,  it  is  convenient  to  arrange  the  terms  so  that 
like  terms  stand  in  the  same  column,  then  to  add  the 
columns  separately  and  combine  the  sums  by  section  50. 


FUNDAMENTAL  PROCESSES  39 

ILLUSTRATIVE  EXAMPLES 

1.  Find  the  sum  of  2a;+3?/  —  4  2,  'Sx  —  i/  -\-2z^  and 
4:x-{-'2y  —  5z,  and  check  the  work  by  letting  a;  =  1,  y  =  1, 
and  2  =  1. 

Solution  Check 

2a:  +  3?/-43  2  +  3-4  =  1 

3x-     y-\-2z  3-1  +  2  =  4 

4a:  +  2y-5z  4  +  2-5  =  1 

9a;  +  4y-72:  9  +  4-7  =  6 

2.  Fin4  the  sum  of  Ta;— 4(?/  +  2),  6  a;  +  2(?/ -f- 2), 
2  a: +(^4- 2),  and  a:  —  3(«/ +- 2),  and  check  the  work  by 
letting  a;  =  2,  y  =  1,  and  2  =  1. 

Solution  Check 

7  a:  -  4(?/  +  z)  14  -  8  =        6 

6x  +  2(2/  +  3)  12  +  4=      16 

2x+    {y  -{-z)  4  +  2=        6 

a:  -  3(y  +  z)  2-6=-    4 

16  a:  -  4(y  +  3)  32  -  8  =      24 

Remark.  Note  that  the  coefficient  of  (y  +  s)  in  the  third  expres- 
sion is  1. 

3.  Find  the  sum  of  ax  +-  hy^  hx  —  ady^  and  cdx  +  ay^ 
and  check  the  work  by  letting  a=2,  i  =  — 1,  <7  =  3,  c?  =  l, 
a;  =  1,  ^  =  1,  and  2  =  1. 

Solution  Check 

ax  -\-hy  2-1=1 

hx-ady  -  1  -  2  =  -  3 

cdx  +  ay 3  +  2  =      5 

(a  +  6  +  cd)x  +(b  -  ad  -{■  a)y  4  -1  =      3 

EXERCISE  18 

Add  the  following  polynomials  and  check  the  results  ; 
1.    a  +  h  2.    a-\-b  —  c 

a  —  h  a        -\-  c 


40  ELEMENTARY  ALGEBRA 

3.    a  —  b  +  2c  4.       a-\-h  —  c  —     d 

a-\-b  —  2c  *2a  —  h+c—'2d 

5.         rr  —  ^  4-  2 
-x+y-z 

7.    3  m  — 2  w 

4  771  H-  3  71 


9. 

-    ab-\-S(^d 

11. 

a—     b  +     c 
-2a+46-6tf 

6. 

x-2y 

X           -2 

8. 

4:X-\-l  y 

-5x-Sy 

10. 

3  m^n  —  8  mn^ 

4  rrv^n  —  3  mn^ 

12. 

^7nn-2m^+    bn^ 

-2  7WW+3m2-ll7l2 

13.    Ja  +  J5  14.        |a  +  ^ 

15.    |aH-|5  16.    _12:4-J«/-J2 

i  a-^-^b  —    x-\-     y-     g 

17.       i^  +  ilZ-i^  18-    .5a;+.3y 

19.    .3  a +  .2  a;  20.    1,2m -{-l.bn 

5a— .Sx  .37/1—    .571 

21.    2w  +  4:v  22.    S7rI^-27rRI£ 

6w-7v  -47ri22+     tt/?^ 

8M;4-2t;  -    ttJ^^  -|-    ttEH 

23.      2^-f-      J?-       O  24.      2a:2-      2:i/+      y2 

3^.-4^4-6(7  -42:2  4.4a;^_2^2 

-5^1  +  4^-6(7  3r»-5a:y  +  3.y2 

25.    4  7w-2w  +  3  26.        3a25-    a5  +  4a^2 

677i  +  5w-l  .    -    a%  +  5ab-Sab'^ 

-2m-4n-4:  2a%-6ab 


FUNDAMENTAL  PROCESSES  41 

27.  m^  -\-6n^         28.  a^  +  3  ab 

-Sm^-4:mn+     n^  -  3  a^  +     ab -\-     b^ 
—  2  mn  —  6n^  6  a^  —  4  J^ 

29.        3(w  +  n)— 3(7w  — Ti)      30.        4(2^  —  3/)+    («^'--v) 

6(m  +  ri)  +    (tw  —  n)  —  6(2;  —  y)—  '^{w  —  v) 

—  2(m4-^)  — 4(y/i  — n)  —  \{x  —  y)  +  8(e^  —  v) 

31.  2  a:  H-  3  ^  -  2,  a;  -  4  y  +  4,  and  3  2;  -  4  ^  —  8. 

32.  3^2  —  2  m^i  —  w^,  4  w^  —  2  wn  +  4  w^,  and 

—  W^  —  4  WW  +  2  7l2. 

33.  3  2;2  +  v^  -  2  ^2^  ^2  _  2  v^  +  t;2,  and  4  v^  -  8  ^2  _  ^2. 

34.  ^_s2_^2^  4r2H-2^^  6  82_2r2,  and4r24.2^2_3^. 

35.  ^Jrx^^-x-\,    3a:2-42;4-4,    "io^-l^-^,    and 
2a:3_^_l. 

36.  a;3_^3^^^^2^        1^-Zxy'^-1y\        4  x^ -{- d  2^y 
+  3  a;^2  _  ^2^  and  3  a^y  —  2  a;^2  _j_  ^3^ 

37.  m^  —  3  7?i2n,  4  7ww2  —  2  w'^,  w^  —  w^,  2  7wri2  —  ^2/1,  and 
n^  4-  3  m7i2. 

38.  3(w  +  w)  +  2(2:-y),  -2(m  +  w)-3(a;-y),   (w  +  w) 
+  (re  —  ^),  and  4(m  +  n)—  2(x  —  y). 

39.  am  +  5w,  6m  +  en,  and  ciw  +  gw. 

40.  ax^  +  6^^  ^^^  —  t?^2,  and  2^2  _  yi 

41.  a6a;  +  w^,  ex  +  wjt?^,  and  ^2;  —  ry. 


42.    4  r  +  3  r2,  Trr  H-  7rr2,  and  cr  + 


ar 


43.  ^22;  +  52^,    —  7i2^  —  c2^,  and  jt?22;  —  y, 

44.  a(m  H-  w)  +  b{m  —  n),      c(m  4-  w)  —  e(m  —  w),    and 
d(^m-\- ri)— hQm  — n). 

45.  a2;  +  52:?/  —  cz,  2  x -\- S  xy  -{■  z,  and  cx  —  xy-{-  az. 

46.  3  2:3  _^  2  2:2^,  3  2^2?/  -  4  ^^2^  5  2:?/2  _  4  ^3^  and  3  ^3  _  2  a^, 

47.  a2:3  ^  52;2  4-  C2;  H-  (^  and  3  2^3  _  2  2^2  ^  3  ^  _  5^ 


42  ELEMENTARY  ALGEBRA 

Subtraction 

53.  Subtraction  of  monomials.  Since  monomials  are 
numbers,  what  is  stated  in  sections  38  and  39  is  applicable 
to  monomials.  Hence,  for  subtracting  one  monomial 
from  another  we  have,  from  section  39,  the  following : 

Rule.  Change  the  sign  of  the  subtrahend  and  add  the 
result  to  the  minuend. 

ILLUSTRATIVE  EXAMPLES 

1.    From  5  m  take  2  m. 


Solution.              5m-(+2w)=5m-2m 

[§40] 

=  3m. 

2.    From  6  a;  take  -2x. 

Solution.               Qx-{-2x)  =  Qx  +  2x 

[§40] 

=  d>x. 

3.    From  —  8  y  take  3  y. 

Solution.          -  83/  - (+  3 .v)  =  -  8;/  -  3  ?/ 

[§40] 

=  -ll.y. 

4.    From  —Qxy  take  —  2  xy. 

Solution.       -  Q  xy  -  {-  2 xy)=  -Q  xy  -{-  2 xy 

[§40] 

=  —  4:xy. 

5.    From  X  take  y. 

Solution.                    X  -(+  y)=  X  —  y. 

[§40] 

6.    From  —  3  xy  take  2  yz. 

Solution.       -Sxy-(+2yz)=-S  xy  -  2 yz. 

[§  40] 

7.    From  —2mn  take  -  4  vs. 

Solution.      —  2  wn  -  (  -  4  rs)  =  -  2  wm  +  4  rs 

[§40] 

=  4  r.s  —  2  mn. 

[§48] 

8.    From  amn  take  bmn. 

Solution.           mnn  -(+  l)tnn)=  amn  -  b7Jin 

[§  40] 

=  (a  —  b)mn. 

FUNDAMENTAL  PROCESSES  43 

EXERCISE   19 

(Solve  as  many  as  possible  at  sight.) 

In   the  first  three  examples  subtract  the  lower  mono- 
mial from  the  upper  monomial. 

1.5a;               — 7a               5w               — 5y  1  z 

2x               —4a               7m               -\-2y  -4z 


2. 

-5E 
-IB 

-%^y 

—     a^             3r^            mn 
-5a2              7^3         4mn 

—  xyz     4(m  +  w)     2(x-\-y) 
xyz     3(w  +  w)     7{x-\-y^ 

-2r8 
-6r8 

3. 

-5(r  +  «) 

4.    From  3  x  take  9  x.  5.    From  4  y^  take  —  7  y^. 

6.    From  —  7  tww  take  2  mn.  7.    From  4  Tri?^  take  irB^. 

8.    From  -  6  7-2  take  -  r2.  9.    From  TriJ^  take  jTri^^. 

10.  From  6{m  +  n}  take  —2(m  +  7i), 

11.  From  (r  -  1)  take  -  2  (r  -  1). 

12.  From  2a(x  +  y^  take  3  a  (rr  +  y) . 

13.  From  52  (a;  _  ^)  take  -2Iy^Qx-y). 

14.  From  0  subtract  x.         15.    From  1  subtract  —  a. 
16.  From  a  subtract  —  b.     17.    From  —  a  take  —  6. 

Subtract  as  indicated : 


18. 

m-  (-11). 

19. 

2^-(  +  y). 

20. 

ax-C-6y 

21. 

a2_(_  ww). 

22. 

c2-(-(?0. 

23. 

_^2_^_^2). 

24. 

—  4a:y  —  (+  Suv). 

25. 

1^^2_(+^2). 

26. 

—  cm^  —  (^—  dm^). 

27. 

ax—  (—  6a;). 

28. 

3(^-y)-«(^-^). 

29. 

w(a;-|-y)-w(a;-f-2^). 

44  ELEMENTARY  ALGEBRA 

54.  Subtraction  of  polynomials.  In  adding  2  a  —  Sb  to 
3  a  4- 5  5  — c,  each  term  of  2  a— 3  J  is  added  to  3a  +  o5  — <?; 
the  resulting  sum  is  5a-{-2h—  c.  Therefore,  in  the  in- 
verse operation  of  subtracting  2a—  Sb  from  5a-\-  2b  —  c, 
each  term  of  2a  —  Sb  is  subtracted  from  ba+  2b  —  c; 
the  resulting  difference  is  Sa-\-  5b  —  c. 

The  work  may  be  arranged  thus  : 

6a  -{■2b  -  c 
-2a  +  36 
Sa  +  5b  -  c 

In  accordance  with  section  39,  the  sign  of  each  term  of 
the  subtrahend,  2  a  — Sb^  has  been  changed  and  the  result 
added  to  the  minuend,  5  a  -\-  2  b  —  c.     In  general. 

To  subtract  one  polynomial  from  another,  change  the  sign 
of  each  term  of  the  subtrahend  and  proceed  as  in  addit/ion, 

ILLUSTRATIVE  EXAMPLES 

1.  Subtract  ^x—2y  from  5  a;  +  3  ?/,  and  check  the  work 
4)y  letting  x—  1  and  y  —  1. 

Solution 

(5a:  +  3y)-(3a:-23/)  =  (5:r  +  3  3/)  +  (-3a:  +  2y) 
=  5  a:  -  3  a:  +  3  2/  +  2  y 
=  2x  +  5y. 

Another  Solution 

5  a;  +  3  ly 

-3a:  +  2y 

2x  +  5y. 

Remark.  In  practice,  the  change  of  the  signs  in  the  subtrahend 
and  the  addition  should  be  performed  mentally.  Thus,  in  the  above 
example  the  work  should  be  arranged  as  follows : 

Solution  Check 

bx+^y  5  +  3  =  8 

^x-2y  3-2  =  1 

2x+  5y  2  +  5  =  7 


FUNDAMENTAL  PROCESSES  45 

2.  Subtract  ^a— 2xi/-{-5b  from  5a  —  35-}-2c,  and 
check  the  work  by  letting  a  =  1,  5  =  1,  c  =  1,  a;  =  1,  and 

Solution  Check 

5a-'db  +2c  5-3  +2  =  4 

3a  +  5&-2xy  3  +  5-2         =6 

2a  -  86  +  2a:y +  2c  2-8  +  2  +  2=-2 

3.  Subtract  bxy  -{-  2bz—  et  from  axy  —  2bz  -{-St^  and 
check  the  work  by  letting  a  =  1,  6  =  1,  c  =  1,  ^  =  1,  2:  =  1, 
and  y  =  1. 

Solution  Check 

aary-2&2  +  3<  1-2  +  3  =  2 

bxy  +  2bz-  ct  1  +  2-1  =  2 

(a  -  b)xy  -  4  62  +  (3  +  c)<  0-4+4  =  0 

EXEBCISE  20 

Subtract  and  check: 


1. 

2x-\-y 
x-y 

2. 

3m4-4n 
2m  +  2n 

3. 

5a  +  35 
2a-    b 

4. 

4m4-3w 

2m-\-Sn 

5. 

3x-Sy 
Sx-9y 

6. 

4a-35 
ba-^b 

7. 

5mn—l 
2mn-\-l 

8. 

5a;2-     X 
la^-h2x 

9. 

27n%2+4 
-3^27^2+5 

10. 

2x^-ly^ 
62:2-8^2 

11. 

-ia-ib 

12. 

4^2-. 4^2 
-    m2-.6/i2 

13. 

bx  +  my 
cx-\-my 

14. 

am^  —  bn^ 

15. 

2irRH+'TrR'^ 
irRH-irR'^ 

16. 

4:x+^y-2 
x-Sy+2 

2m  +  Sn  —  4p 
Sm^-in—  5p 

17. 
19. 

a  + 
—  a  — 

b  +  e 
b-e 

18. 

4r-h 

\8-6t 
8-\-  5t 

46  ELEMENTARY  ALGEBRA 


20.    -3a;2  +  2a;-4 


22. 

4  77^2  4.  2  m  +  5 
m2            -1 

24. 

2r  +  « 
-5*+    r-8 

26. 

x^-    y 
4:x-\-2y-bz 

-3 

28. 

3  a  +  3(m  +  w) 
a  —  2(m  4-  w) 

21. 

23. 

^+      y  +  2 

5a:-4^/ 

52^            -3 
2a^  +  4a:-5 

25. 

am  -{-  S  bn  -\-  4:  dx 
cm  -\-     hn  —  ^dx 

27. 

—  m  —  b  n  —     p 

—  Sm            —2p- 

-3 

29. 

-2(a  +  i)-2 

30.    2(a;  +  ^)+3(m+w)         31.    _3(a+ ^)2-4(c-(i)2 
4(a;  +  y)+2(7/^  +  ^)  2(a+^)^-4(g-(^)2 

32.       a  +  5  -  c  -  (^  33.      (a  +  hy  -z-^S 

Sa-b-c-^d  2(a  -|-  5)^  +  g  -  2 

34.  5(10)2 +  6(10) +7 
2(10)2 +  4(10) +8 

35.  7.68-3-62+2.6  +  1 
6.63-8-62-3.6  +  4 

36.  From  a  +  b  take  3  a  +  4  5  —  c?. 

37.  From  Sm^  -\-4:m  —  n  take  4  9^2  —  ^2  +  w. 

38.  From  ax^  -{-a^x-h2  a^aP  take  Sa^x^-S  aH  +  2  ax^, 

39.  From  a  -  1  take  ^2  +  a2  +  a  +  2. 

40.  From  x-^-7?'  take  5  rr^  _  3  ^  9  a:. 

41.  From  w  +  2(a  +  6)  take  3(a  +  J)  -  2  w  +  4. 

42.  Subtract  4  a;^2  __  3  ^y  _|_  ^.^  from  xy. 

43.  Subtract  3  r^  -  a;  +  4  from  0. 

44.  Subtract  2  a»  -  3  a  +  4  from  3  a"  -  2  a  +  4. 


FUNDAMENTAL  PROCESSES  47 

45.  Subtract  la^-^a-^-b-^  from  4  +  -i-  5  +  a2  _,.  i  ^, 

46.  From  ax^  —  bi/^  -\-  cz^  take  ma^  —  my^  +  2  s^. 

47.  From  rr^—^T^  —  'p^  take  a/^i^ _  5^2 _  ^^^2 

48.  Take  3  m^  —  ti  —  4  from  the  sum  of  vr^  —Zm-\-\n 
and  3  m^  —  m  —  6. 

49.  Take  1  -^  r^  —  ^  -\-  r%  from  the  sum  of  3  r^  4-  r«  and 

50.  Take  the  sum   of  a^  —  ^a-\-  2  and    —  a^  —  ^  a^  -\-  a 
from  1. 

51.  Take  a^  -\-  x^  —  x  -\- 1  from  the  sum  of  .r^  —  x^i/  H-  a:^ 
and  —  a:^  —  xt/^  +  1. 

52.  From   the   sum    of   wB^  and  lirRH+'lirB?  take 
4  7ri22  _  ^RH. 

53.  From  the  sum  of  a  +  b  —  c  and  2«  —  35+2c  take 
the  sum  of  —  a-^  2h  and  2a  —  36  +  2  c. 

54.  Take    the  sum   of   32:  —  4^  +  5  and   3  y  —  4  a:  —  4 
from  2. 

55.  From  x-\- 1/  -[-z  subtract  the  sum  of  a:  —  2  y  —  z  and 
2x-y-\-2z. 

If  A  =  Sx-\-2y-5z%   B  =  2x-Zy^4:z\   and    (7  = 

—  X  -{■  y  +  z\  find  the  value  of  : 

56.  A^B-0,       57.    A-B+C.       58.    B-^C-A. 
59.    A- B-0,  60.    B-C- A. 

61.  From     3  a^hc  -  2  ab^  -\-  5  b^c     take     2  a2^c  +  2  ^^2 

—  3  62c  +  2  flic. 

62.  From     4  xyz^  —  2  a;!/22  -|-  7  a^?/3     subtract     —  5  xyz^ 

—  13  2:2/^2  +  a^yz. 

63.  From     22(a  +  6)2  +  5(«  +  6)- 7     take     (a  +  6)2 
-5(a  +  6)4  3. 

64.  From     5(2;  -^  y)+  13(a  +  6)  -  2     take     7(a  +  6) 
-(2:  +  ^)+2. 


48  ELEMENTARY  ALGEBRA 

65.  From  x  +  (^  +  zy  +  t^  take  6x  +  t/-^l, 

66.  How  much  greater  than  x-\-l  is  x^? 

67.  How  much  less  than  a  +  35  —  4<?is  2  a-{- b  -\-  c^l? 

68.  What  must  be  subtracted  from  a  so  that  the  re- 
mainder is  h? 

69.  What  must  be  subtracted  from  x'^  —  xy  -\-  y^  so  that 
the  remainder  is  2  a:^  _}_  3  v 

70.  From  the  sum  of  W  xy^  —  \%  :)iP'yz -\- VI  xyz  and 
3  xyz  —  2  Tp'yz  -{-  b  x^  —  S  y  take  the  sum  of  5  xyz,—  13  xyz^ 
+  12y  -19x^  and  7  x^yz  -lSx^-\-  y. 

71.  From  ax^  +  bxy  +  cy^  take  J  aa^  _  1. 5^^  _|.  2  cy^. 

72.  Take  the  sum  of  Sh^-2hk  +  Sk^-^2h-^k-\-l 
and  4A;2-3^^+57i  +  3^  +  2  from  2h^  +  2hJc-nJ(^ 
4- 11  A -16  A; +7. 

Parentheses 

55.  2  a  4-  (3  6  -H  c)  means  that  the  number  (3  6  +  c)  is 
to  be  added  to  2  a. 

2  a  +  (3  6  —  ^)  means  that  the  number  (3  6  —  c?)  is  to  be 
added  to  2  a. 

By  addition, 

2a-f(3  5-hO=2«  +  3J  +  c; 
and  2a-}-(3  6-0=2a+35-c. 

Therefore, 

1.  If  an  expression  in  parentheses  is  preceded  by  the  plus 
sign^  the  parentheses  may  he  omitted. 

2.  An  expression  may  be  inclosed  within  parentheses  pre- 
ceded by  the  plus  sign. 

Thus, 

1.  2  a  +  (3  6  -  c  +  ^)  may  be  written  2a  +  3A-c  +  (f. 

2.  2  a  +  3  ?>  -  c  +  rf  may  be  written  2a+(3  6-c  +  d),  or 
2a +  36+(- c  +  c?). 


FUNDAMENTAL  PROCESSES  49 

2  a  —  (3  6  H-  e)  meaus  that  the  number  (3  6  +  <?)  is  to  be 
subtracted  from  2a.  2a  —  (— 5  —  <?)  means  that  the 
number  (  —  5  —  c)  is  to  be  subtracted  from  2  a. 

Since  the  result  of  subtracting  a  number  is  the  same  as 
the  result  of  adding  the  opposite  number, 

2a-(3  6H-(?)=2a  +  (-3  6-c)  =  2a-36-c; 
and  2a-(-6-c?)=2a4-(6-|-c)=2a+6  +  c. 

Therefore, 

1.  If  an  expression  in  parentheses  is  preceded  hy  the 
minus  sign^  the  sign  of  each  term  within  the  parentheses  must 
he  changed  when  the  parentheses  are  removed. 

2.  An  expression  may  he  inclosed  within  parentheses  pre- 
ceded hy  the  minus  sign,  provided  that  the  sign  of  each  term 
be  changed. 

Thus, 

1.  2 a  —  (3 6  +  c)  may  be  written  2a  —  Sb  -  c. 

2.  2a  -  3b  +  c  -  d  may  be  written  2a-(Sb-c  +  d)  or  2  a 
-Sh-(-c  +  d). 

Remark.  Observe  that  when  the  parentheses  preceded  by  the 
minus  sign  are  removed,  the  minus  sign  before  the  parentheses 
is  omitted. 

ILLUSTRATIVE  EXAMPLES 

1.  Simplify     2 +  [3 -(5 -2)]. 

Solution.     2+[3 -(5-2)]  =  2+[3- 5  +  2] 

=  2  +  3  -  5  +  2,  or  2. 

2.  Simplify     2a -[3 a- (2 a- 5)]. 

Solution.     2a-[3a-(2a-Z>)]  =  2a-[3a-2a  +  6] 

=  2a  —  3a  +  2a  —  6,  ora  —  6. 

3.  Simplify  1  -  [2  a:  +  J3  y  -  (4  «  +  5)  j]  +  [5  a: 
-S4^4-(32-2)S]. 


50  ELEMENTARY  ALGEBRA 

Solution.     l-l2x  +  {3y  -(4:z  +  5)}]  +  [5a:  -  {4y  +(32  -  2)}] 
=  1-  [2a: +  {3  2/ -42-  5}]  +  [5x  -  {4:y  +  Sz  -  2]'] 
=  l-[2a:  +  3y-42-5]  +  [5x-42/-3z+2] 
=  l-2a;-3y  +  4z  +  5  +  5a:-43/-32  +  2 
=  S  +  3x-7y  +  z, 

Remark.  It  will  be  observed  in  the  solutions  of  illustrative 
examples  1,  2,  and  3,  that  the  innermost  parentheses  have  been  re- 
moved first.  Although  it  is  not  essential  to  do  so,  yet  the  beginner 
is  advised  to  proceed  in  this  manner. 

4.  Indicate  that  from  the  sum  of  2a-\-Sb  —  c  and 
Sa  +  b-\-2c  the  sum  of  a  —  h-\-c  and  4a+-26  —  3(?  is  to 
be  subtracted ;  then  perform  the  indicated  operations. 

Solution. 

[(2a  +  3&-c)  +  (3a  +  &  +  2c)]-[(a-&  +  c)  +  (4a  +  26-3c-)] 
=  [2a  +  36-c  +  3a  +  &  +  2c]-[a-ft  +  c  +  4a  +  26-3c] 
=  [5a  +  46  +  c]-[5a+  />-2c] 
=  5a  +  46  +  c  -  5a  -  6  +  2c 
=  3  &  +  3  c. 

EXERCISE  21 

(Solve  as  many  as  possible  at  sight.) 
Remove  the  parentheses  and  simplify  when  possible  : 

2.  a:+-(y-3). 

4.  m—{n—p). 

6.  x-\-(2x-y). 

8.  5 +  (7 -2). 

10.  8 -(5 -4). 

12.  ^x  —  (2x  —  x). 

14.  r— (— 3r-fr). 

16.  2k  — [m  —  w] . 

18.  9-7"=^. 

20.  7 -(8 -5)- 2. 

22.  6-(5-l)-(4-3). 


1. 

a-\-(h  +  c). 

3. 

x-iy-^z). 

5. 

r-(B-ty 

7. 

a-(a-\-b). 

9. 

6 -(3 +  2). 

Il- 

3-(-2-l). 

ia. 

2m-(-3m-m). 

15. 

7rH-!-2«-«S. 

17. 

8-5+2. 

19. 

9-(64-2)  +  3. 

21. 

8-(6  +  l)-(8-7) 

FUNDAMENTAL  PROCESSES  51 

23.  a-(a-^)  +  (a- 25). 

24.  2x-(x  —  i/}  —  (2x  +  2i/), 

25.  2^-(32^-6i/)-(z^-2y  +  3y). 

26.  4a:  +  [3a: +(3  2: -2)]. 

27.  7a:- [3a:-(6«/  +  22)]. 

28.  2m  +  37i  +  fm  —  (m— n)}. 

29.  a:-2^-52a:-(a:-3^)S. 

30.  6r+ J37--(2r-0-3«i. 

31.  4m— [2w  — (3m  — n)-|-2  7i]. 

32.  (3r  +  s)-{r-(2r-«)  +  3r|. 

33.  (4^-^)-[-;>-(3^  +  ^)-4^]-(;>  +  ^). 

34.  a:- [a:-(«/  +  2)-Ja:-(^-2)i]. 

35.  ll_J10-[9-(8-7-6a:)]i. 

36.  2-S-2-[2-(-2-2T2-2)-2]i. 

37.  r  —  \_—  r  —  I  —  r  —  (r  -\-  r  —  r  —  r  —  r}  —  r  \  —  r^. 

In  examples  38-44,  remove  only  the  inner  parentheses 
and  simplify  when  possible. 
Thus,  [a  -  (6  -  c)]  =  [a  -  &  +  c]. 

38.    [a:-(^/  +  2)].  39.    [a:  +  (y  +  2)]. 

40.    [m  — (w  — 1)].  41.    [{rn  +  n)  —  (p  —  q)], 

42.  [(2a:+3^)-(2a:-3y)]. 

43.  [(a:2_^2)_(^+2./2)]. 

44.  5  (m  +  n)  —  (m  —  7i)  j . 

In  examples  45-50,  indicate  the  operations  before  per- 
forming them. 

45.  To3a2_ladda2_3a-Hl. 

46.  From  3  a  take  a  —  2. 

47.  Add  7^-2o?y,  3  xy^  -  y\  and  a:^^  -  2  xy^. 


52  ELEMENTARY  ALGEBRA 

48.  From  3  y  take  ^  —  5. 

49.  Add  1  to  the  sum  of  3  a  —  3  and  2  a  +  2. 

50.  Take  the  sum  of  3a  —  26  +5  and  6  —  5a  —  2  from  2. 

Inclose  the  last  three  terms  of  the  following  polyno- 
mials within  parentheses,  preceded  by  a  plus  sign. 


51. 

a-\-h-\-c  -\-d. 

52. 

—  a-^h  —  c-{-  d. 

53. 

a-\-h-\-c—d. 

54. 

a-{-b  —  c  —  d. 

55. 

m  — w+jp  +  l. 

56. 

m  —  n—p  —  1. 

57. 

2^2  +  ^2  _,.22/  +  l. 

58. 

m^  —  n^  —  2  n  —  1, 

59. 

r2_82^2«-l. 

60. 

x^  —  y^  —  2  yz  —  z^. 

61-70.  Inclose  the  last  two  terms  of  the  polynomials 
in  examples  51-60  within  parentheses,  preceded  by  a  plus 
sign  when  the  sign  of  the  third  term  is  plus,  and  by  a 
minus  sign  when  it  is  minus. 

71-80.  Inclose  the  last  three  terms  of  the  polynomials  in 
examples  51-60  within  parentheses,  preceded  by  a  minus 
sign. 

EXERCISE  22.  — GENERAL  REVIEW 

1.  Simplify  by  collecting  like  terms  : 
2x—Sy-\-z  —  i/-^Sz-{-2x  —  Sx  —  y-^4:y  —  i2, 

2.  Evaluate  a[^—2x^  —  x-\-l  when  x  =  1. 

3.  Evaluate  a^  —  4:  x^  -\- x  -\- 1  when  x=2. 

Given  7r=  3.1416,  find  to  three  places  of  decimals  : 

4.  2  7rr,  when  r  =  1.  5.    2  7rr,  when  r  =  1.5. 
6.    7rr,  when  r  =  2.                   7.    7rr,  when  r  =  2.5. 
8.    4  Trr^,  when  r  =  4.              9.    ^  Trr^  when  r  =  3. 

10.  Express  the  sum  of  the  squares  of  a  and  h;  the 
difference  of  the  squares  of  c  and  d. 


FUNDAMENTAL  PROCESSES  63 

11.  If  m  and  n  represent  two  numbers,  what  does 
m-\-n  represent?  What  does  m  —  n  represent?  m^  -\-n^'^ 
TT^  —  r^^.     (771  +  w)2  ?     (m  —  7i)2  ? 

12.  Express  the  sum  of  the  cubes  of  x  and  y\  the 
difference. 

13.  Simplify  by  collecting  like  terms  : 

3  a26  +  2  aJ  -  a^^-  2  a%  -  3  ab^  -  2  a% -\- 2  ab -^  4:  ab\ 

14.  Add    a;  +  2y4-3z,    2x  —  y  —  2z^    y  —  x—  z^    and 

15.  Add  a4-2a3_^3a2,  a  +  a^ -\- a\  2a^4-3a3,  a2+5a 
-  2,  and  -  2  r-  3  a  -  4  a2. 

16.  Add  -2(x-y),  6(x-y),  -4(a:-2/),  and  7(a:-y). 

17.  Simplify  by  collecting  like  terms : 

^x-  5(y  -  2)  —  2  a;  +  (^  -  z)  +  2;  -  2(y  -  0)  4-  2  2. 

18.  From  5  a  take  2  a  —  3. 

19.  From  3  take  x^—  x  —  1. 

20.  From  a^ -\- 2  ab -\- b^  take  a^-2ab-\-  b^. 

21.  From  2  take  the  sum  of  2a— 3^  —  4  and  2  —  2b-\-  a. 

22.  From  3  a6c  -  2  a%x  -\-l  ay-2  take  -  3  a6c  -  3  a25a; 
+  7  ay  +  2. 

23.  What  must  be  added  to  m  —  w  +  jt?  to  make  2  jp  ? 

24.  What  must  be  added  to  a:  +  y  —  z  to  make  0  ? 

25.  What  must  be  subtracted  from  a-\-b  +  c  to  make 
a-b-\-cl 

26.  If  2  «  =  a  +  6  +  c,  what  expression  is  equal  to 
aH-6-c? 

27.  Given  «  =  (a  +  5)  ^ ;  for  what  number  does  «  stand 
when  a  =  1,  ft  =  3,  and  w  =  6  ? 


54  ELEMENTARY  ALGEBRA 

28.  Simplify  by  combining  the  terms  having  the  same 
powers  of  m,  so  as  to  have  the  plus  sign  before  each  set  of 
parentheses : 

am^  +  Jw2  +  <?w  —  3  w^  —  4  m  —  7  —  5  m^. 

29.  Simplify  by  combining  the  terms  having  the  same 
powers  of  x^  so  as  to  have  the  minus  sign  before  each  set  of 
parentheses : 

30.  If  ^  =  2a:2-3a:+l,  B^x^-^x-'l,  and  (7  = 
rr  -  4  +  0^2^  find  the  value  of  ^  +  ^  +  6\ 

31.  Evaluate  Sa;  —  [4^  —  (2^;  —  y  —  Z  y^  —2x']  when 
x  =  l  and  y=2. 

32.  Subtract    2(x  -  y^ -(x  -  y)-{-l    from    4(a:  -  y)2 

33.  If  A=  ax-{-2by  —  cz  and  B  =  x—  Zby  —  kz,  find 
the  value  of  ^  —  J5. 

34.  If  A  =  m(ix  —  y^ -^  r(^x  +  y^  and  B  =  n(x  —  y) 
—  (a;  H-  y),  find  the  value  of  ^  —  ^. 

Multiplication 

56.  Commutative  law  for  multiplication.  The  product 
of  a  and  b  is  expressed  by  ab.  Similarly,  the  product  of 
a,  6,  and  c  is  expressed  by  abc. 

In  arithmetic  it  is  obvious  that 

3x5  =  5x3. 

In  algebra  it  is  assumed  that : 

The  product  does  not  depend  on  the  order  in  which  the 
factors  are  taken. 

Thus,  it  is  assumed  that 

axb=h  X  a. 

This  important  principle  is  referred  to  as  the  commuta- 
tive law  for  multiplication. 


FUNDAMENTAL  PROCESSES  55 

57.  Associative  law  for  multiplication.  In  arithmetic  it 
is  obvious  that 

2x(3x5)  =  (2x3)x5. 
That  is,  2  X  15  =  6x5. 

In  algebra  it  is  assumed  that: 

The  product  does  not  depend  on  the  way  in  which  the 
factors  are  grouped. 

Thus,  it  is  assumed  that 

a  X  (6  X  c)  =  (a  X  6)  X  c. 

This  important  principle  is  referred  to  as  the  associative 
law  for  multiplication. 

58.  Index  law.     From  section  14,  we  have 

23  =  2  X  2  X  2 ; 
and  2*  =2x2x2x2. 

Therefore,  2^  x  2*  =  (2  x  2  x  2)  x  (2  x  2  x  2  x  2) 
=2x2x2x2x2x2x2 
=  27,  or  23+^ 
Similarly,    a^  ^  a*  =  (a  x  a  X  a)  X  (a  x  a  x  a  x  a) 
=axaxaxaxaxaxa 
=  a^,  or  a3+4. 

In  general,  when  m  and  n  are  positive  integers, 

2%e  exponent  of  the  product  of  two  powers  of  the  same  base 
is  equal  to  the  sum  of  the  exponents  of  the  factors. 

Remark.  Observe  that  any  power  of  a  positive  number  is  positive 
and  that  all  even  powers  of  negative  numbers  are  positive,  while  all 
odd  powers  of  negative  numbers  are  negative. 

For  example, 

(-  a)2  =(-  «)(-  «)=  a2  and  (-  «)»  =(-  a)(-  a)(-  a)  =  -  a*. 


56  ELEMENTARY  ALGEBRA 

EXERCISE  23 

Name  the  product  in  the  following  examples : 


1. 

x'a^. 

2. 

m^m. 

3.     1/2^6. 

4. 

r^rK 

5. 

sh. 

6.    tK'. 

7. 

-  aHK 

8. 

a(-a6). 

9.     w2(  —  7W^). 

10. 

22 .  23. 

11. 

3.32. 

12.    TT  X  7r2. 

13. 

Rm. 

14. 

a'^a. 

15.    a'^a*. 

16. 

x-^x\ 

17. 

Cmf^m^ 

18.    a;2a^. 

19. 

-x'^x. 

20. 

-  x'xK 

21.    a:(-a:*»). 

22. 

a?"^a^'^. 

23. 

—  x'^o?"^. 

24.    r(-y)- 

25. 

c-an- 

a3). 

26. 

(- 

-«)2(-a)3. 

27. 

(-«')(- 

a8). 

28. 

(^ 

^+y)'(^+^). 

29. 

(x~\-i/y(ix-{-yy. 

30. 

ix 

+  y-2)(a;-|-y-2; 

31. 

a^  '  a^  -  a^. 

Suggestion,     a?-  •  a^  '  a^  —  {a^a^^a!^  —  a^a*  =  a^. 

32.  (iab)\ahy(aby.         33.  (x-^i/yix+i/y(x-\-yy. 

34.  (a  +  ^  -  c)(«  +  ^>  -  cy(a  +  b-  cy. 

35.  (m_  1)2(^-1)3(^-1)7. 

36.  (2x-  ^ yyC2x  -  S  2/y(2x  -  S yyC2x  -  S  I/). 

59.   The  product  of  two  monomials.     When  written  in 

full, 

2abxSab^='2xaxbxSxaxbxb 

z=2xSxaxaxbxbxb  [§  5^] 

=  (2  X  3)  X  (a  X  a)  X  (6  X  6  X  ^>)  [§  57] 
=  6  a253. 

In  like  manner  it  may  be  shown  that 

(  -  2  a^2^)(  __  4  ab^c)  =  8  a%^(^. 
Also,  that 

(_  2a62)(-  3a262)(-  4a5)  =  -  24 aW 


Euclid  (330  275  bc)  was  a  successful  teacher  of  mathematics 
in  Alexandria.  His  Elements  has  been  the  recognized  textbook  in 
elementary  geometry  for  2000  years.  In  it  are  to  be  found  geo- 
metrical proofs  of  the  commutative  and  distributive  laws. 


FUNDAMENTAL  PROCESSES  57 

From  the  foregoing  illustrations  it  is  evident  that : 
The  product  of  two  or  more  monomials  is  equal  to  the  prod- 
uct of  their  numerical  coefficients  and  all  the  different  literal 
factors  that  occur  in  the  monomial  factors^  each  letter  having 
as  exponent  the  sum  of  the  exponents  of  that  letter  in  the 
monomial  factors. 

EXERCISE  24 

Multiply : 

1.  2  a  by  a.  2.  2  jK  by  ^.  3.    -  3  a;  by  4  x. 

4.  -Qyhy  -2y.  5.  3a^  by  2a:3.  g.    ^Rhy^R. 

7.  a%'^  by  a%^.       8.  6  a  by  -  3  6.        9.    2  i2  by  J  i. 

10.  -  4  xyz^  by  -  2  A.  ii.    4  22  by  ^  i2. 

12.  I  a%  by  -  3  he.  13.    -2ah'^hy  -  3  ax^. 

14.  -  ?>a%c  by  4  he^x,  15.    (a  +  hy  by  3(a  -h  6)3. 

16.  2(x  +  yyhy  -^(x-^yy. 

17.  -  4  <a  + 5)2  by  3  2:2(^^^)8. 

18.  —^a(x-\-  yY  by  -  2  a2(2:  -f  yy, 

19.  3(2:  +  y)3by  (a: +  3/). 

20.  -  3(a;  +  yy  by  -  (2;  +  y)^. 
Perform  the  indicated  multiplications: 

21.    (Za%'^y.  22.    {-2abe^y.        23.    (fa;)^ 

24.    (-fa3)2.  25.    (-|a26)2.  26.    (-3a2^,3^)2. 

27.  (a62)(3a252)(2a25). 

28.  (3mw2)(_2m2n3)(-4  7W7i). 

29.  (-2:3/)(32^./3)(_5^4^). 


30. 

(2^25)(_|«J)(2a358). 

31. 

(-y)(-/)(-^). 

32.    (2a26)3. 

33. 

(-3^)3. 

34.    C-^axy. 

35. 

(-.4a263c2)8. 

36.    (-Ja262^)8. 

68  ELEMENTARY  ALGEBRA 

37.  b(x  +  yX^  +  yy(x  +  yy. 

38.  2(a-hl)2(a  +  l)3(a+lj)4. 

39.  -2>(x  +  y-{-  z)\x  ^y  +  z)\x  -\- y -\-  zy>. 

40.  3a(a;-h  «/)  •  2a5(ajH-^y)  .  3  5c(a7H-y). 

41.  (  -  3  a%c)  (f  a52)  (  -  4  o^c^)  (  -  ^253^) . 

42.  (1 0^33^2254)  (  _  2  2:2^23)  (  _  3  ^^3^2)  (  _  ^^2^5) . 

43.  3  a;"*.  4  a;".  44.    5  ^''.4^. 

45.    2  a"» .  3  a2*".  46.    5  a;"'^!  •  3  rc*""^. 

47.    3  af +^  .  2  a;«-*.  48.    -  3  a^^^C  -  Ta:^). 

49.      _7^2o+l(_3^2-2a),  50       (_3a;".)2. 

Find  the  product  of  : 

51.  a;,  3  a;2^  2  x",  and  4  a;^ 

52.  2  a26,  6  62^,  and  5  c^a. 

53.  -  3  hV,  5  H2,  and  -  7  ^3^2. 

54.  28,  -  32^,  -  4,  and  52^3. 

55.  ^x^yz^  —  ^xy\  ^xyz^^  and  —  2xyz. 

56.  |(a:+  y^{y+zy,  -  J(y  +  2!)(25  +  a;)2,  and 
-K2  +  a:)(a;  +  ^)2. 

60.  Multiplication  of  a  polynomial  by  a  monomial.  It 
is  obvious  that 

3x(4  +  5)=3x4  +  3x5. 
Also,  that 

3  X  (4  +  5- 2)=  3  x4  +  3x5-3x2. 

In  like  manner,  it  is  assumed  that 

a(h-\-c)-=ah-\-ac.  (1) 

Also,  a(J>-\-c-d)  =  ah  +  ac-ad.  (2) 

Equations  (1)  and  (2)  express  the  fact  that  multiplying 
every  term  of  a  polynomial  hy  a  monomial  multiplies  the 
polynomial  hy  that  monomial.  This  principle  is  referred 
to  as  the  distributive  law  for  multiplication. 


FUNDAMENTAL  PROCESSES  59 

From  equations  (1)  and  (2)  we  have  the  following  rule 
for  multiplying  a  polynomial  by  a  monomial  : 

Rule.  Multiply  each  term  of  the  polynomial  hy  the  mono- 
mial and  write  in  succession  the  resulting  products^  each  with 
its  proper  sign. 

ILLUSTRATIVE  EXAMPLES 

1.  Multiply  (h-\-  c—  d^  by  2 a  and  verify  the  result 
when  a  =  2,  5  =  3,  (?  =  2,  and  d  =  l. 

Solution.         2a(h  +  c  -  d)=2ah  +  2ac  -2ad. 
Check.        2x  2(3 +  2-1)=  2x2x3  +  2x2x2-2x2x1 
4x4  =12+8-4. 

16  =  16. 

2.  Multiply  (-2ar*  +  3y-222)  by  ^x^yz  and  verify 
when  x=  —  2,  y  =  —  1,  and  2  =  —  3. 

Solution  Check 

-2a:2  +  3y-222  _  8  -  3  -  18            =      -29 

3  x'^yz 36 =  36 

-  6  x^yz  +  9  x'^yH  -  6  x^y^  -  288  -  108  -  648  =  -  1044 

Remark.  The  form  of  solution  used  in  example  2  is  preferable 
when  the  multiplication  cannot  readily  be  performed  at  sight. 

3.  Simplify  2(a  -  26)+  3(2  a  -  5). 

Solution.        2(a  -  2 fe)+  3(2  a  -  ft)  =  2  a  -  4  6  +  6  a  -  3  6. 

=  Sa-7b, 

EXERCISE  25 

(Solve  as  many  as  possible  at  sight.) 
1.    a(x+-y).  2.    a;(a  — 6). 

3.    2'irR{H-\-R).  4.    y(a-6  +  c). 

5.    x(x  —  y  —  z),  6.    —  a(— a  — 6  — c). 

7.    ah{a  —  h  —  V).  8.    Q[^{xy—xz-\-c). 

9.    7r(^  +  r  -\-  Rr).  10.    m\mhi  —  m^p  +  pn). 


60  ELEMENTARY  ALGEBRA 

11.  _2(3a-45  +  c).  12.    6(2^-22/3-322). 

13.  2ax(ax-l  +  Sa^3^),  14.    -  2a2«/(12a3  +  3y). 

15.  Say^-x-^lhy  ^x.  16.    2a:  + «/+ 1  by  -a;. 

17.  3a:2  -  a:  +  2  by  3  x\  18.    2aft2  -ah+h'^  by  aJ. 

19.  a2_3^5  4.262by_2a5. 

20.  ^x^y  —  2xy^  —  ^y^  by  xy^, 

21.  ah  -\-hc—  ea  by  —  aftc. 

22.  ?;2  _  ^3  _  4  ^4  |3y  ^^^ 

23.  a(a;  +  «/)  +  (wi  +  7i)  by  ab. 

24.  a(m  +  ?2)  + ^(jo  + 5')+ <?(a^  +  ^)  by  aic. 

25.  (a:4-y)-(a:  +  ^)2  +  (2J  +  «/)3by  (rr  +  3/)2. 

26.  2  a(w  +  7i)  —  3  5(m  +  /i)^  +  4  <?(m  +  n)*  by  (m  -\-  w)2. 

27.  4(a;  H-  «/)2—  3 ^(2;  +  ?/)  —  5 c(rr  +  i/)^  by  2 a6c(2;  +  «/)*. 

28.  a^(^x  +  ^)  —  ^^(^  +  «/)^  —  abc(x  +  ^)3  by  abc(^x  +  ^)2. 

Simplify  : 

29.  3(a  +  26)+2(a-6).         30.    (2  -  a:)y- a;(l  +  y). 

31.  (22  —  3^)2:2 +  (3y  —  42)^2  —  (22  — 3a:)2y. 

32.  a[l +(^ +<?)]• 

33.  [a  — 2(a;— a)]^;  — [a;+2(a  — a;)]a. 

34.  5a3_2a(2a2_a  +  l)-3a(a2-2a  +  l). 

35.  3(1  +  20  +  300),  36.    5(2  +  3a:-2^  +  |2). 
37.  a»(l  +  a  +  a2).                38.  x^'^x'^y'^-^- ^x^y""  -  ^}, 

39.  a'"a;«(  -  2  aa;2  +  3  a"»+ V+2  _  7  a;"+3) . 

40.  -3m27lP(-fm«7l2P4.|t7^9+lw3P+l-|). 

41.  (2:  -  yyiCx -  yy  -  3(2:  -yy  +  2(x-y)'\-il 

*'•  "L"34 51- +  -68-] 

43.  7a»  -  2a(a2  +  3a  -  2) -  SaCa^  -  2a  +  5). 


FUNDAMENTAL  PROCESSES  61 

44.  To  the  product  of  3  a -|- 1  and  5  a  add  3  a  — 2,  and 
multiply  the  sum  by  a. 

45.  From  the  product  of  3m2-j-3mH-2  and  bm  sub- 
tract the  product  of  5 m^  —  2m  —  1  and  3m. 

61.  Multiplication  of   a    polynomial   by  a  polynomial. 

The  product  of  two  polynomials  can  be  obtained  by  suc- 
cessive applications  of  the  principle  given  in  section  60  ; 

thus  : 

(3  a  -  4  6)  (a;  -  2/)  =  (3  a  -  4  6) a:  -  (3  a  -  4  &)3/ 
=  (3  aa:  -  4  hx)  -  (day  -  ^hy) 
=  3  ax  —  4:  bx  —  S  ay  +  4:  by. 

In  like  manner  : 

(3  a  _  4  J  +  2  c)(2  a  -  3  6)  =  (3  a-4  &  +  2  c)(2  a)  -  (3  a-4  b  +  2  c)(3  b) 
=  Qa^-Sab-\-4ac-9ab  +12b^-Qbc 
=  6  a2-17  a6  +  4rtc  +  12  b^-Q  be. 

Remark.  From  section  60,  z(x  —  y)=  zx  —  zy^  in  which  expression 
the  letter  z  represents  any  number.  Substituting  (3  a  —  4  6)  for  z, 
we  have 

(3  a  -  4  6)  (a:  -  y)  =  (3  a  -  4  6)x  -  (3  a  -  4  b)y, 

which  is    the  expression  obtained    in  the  first   of    the  preceding 
examples. 

From  the  foregoing  illustrations  we  have  the  following 
rule  for  multiplying  a  polynomial  by  a  polynomial  : 

Rule.  Multiply  each  term  of  the  multiplicand  hy  each 
term  of  the  multiplier  aiid  add  the  partial  products. 

Note.  Before  multiplying  one  polynomial  by  another,  both  of 
them  should,  if  possible,  be  arranged  according  to  the  ascendinr/  or 
descending  powers  of  a  certain  letter ;  that  is,  in  such  a  manner  that 
the  exponents  of  a  certain  letter  in  successive  terms  decrease  or 
increase  from  left  to  right. 

For  example,  3  ar^  —  a:"  -f  4  a:  —  3  is  arranged  according  to  the 
descending  powers  of  x ;  and  a  —  ay  -\-  2by^  —  y*  is  arranged  accord- 
ing to  the  ascending  powers  of  y. 


62  ELEMENTARY  ALGEBRA 

ILLUSTRATIVE  EXAMPLES 

1.  Multiply  3a^-f2a;— 1  by  2a;— 3  and  verify  when 

a;=  2. 

Solution  Check 

3a:2  +  2a:-l  =15 

2x  -3 =1 

6  a;8  +  4  a:2  -  2  a; 

-  9  ar^^  -  6  ar  +  3  __ 

6a:«-5a:2-8ar  +  3  =15 

2.  Multiply  a— 5-f<?bya-f6  and  verify  when  a  =  5, 
5=3,  and  c  =  2. 

Solution  Check 

a  —  ft  +  c  =4 

g  +  & =    8 

+  gfe  -  fc''  +  ^>g  

a2  +  ac  -  fe2  ^.  2,c  =32 

3.  Multiply  3  a:^  —  2  xy^  +  3  x^y  —  y^  by  —  y^  -\- xy  •\' a^ 
and  verify  when  a;  =  2  and^  =  1.    (See  note,  section  61.) 

Solution  Check 

3  a;8  +  3  x2y  -  2  xy^  -  f  =31 

x^  -\-  xy  —  y^ =      5 

3  a:«  +  3  a;*y  -  2  T^y^  -     x^y^ 

+  3  a:*3/  4-  3  x^y"^  -  2  xh^  -  xy^ 

-  3  x^y^  -  3  x^y^  +  2  xy*  +  y^  

3  a:*  +  6  xV  -  2  a;8y2  _  6  ara^s  +  ^,^4  _^  yi        =  155 

EXERCISE  26 

Multiply,  and  check  results  : 

1.    a; +  3  by  a: +  2.  2.  2wH-lbyw  +  3. 

3.    a  +  5by2aH-4.  4.  3  a; +  5  by  a; -4. 

5.    5  a  +  2  by  2  a  +  3.  6.  2  m  -  4  by  3  w  -  3. 

7.    6-4a;by  5-2a;.  8.  3a -5  by  4  + 6a. 


FUNDAMENTAL  PROCESSES  63 

9.  5  +  2  ?w  by  3  w  -  2.        lO.    8  a  +  2  by  3  a  -  2. 

11.  4  —  2  ^  by  4  4-  2  y.  12.    m  -\-  n  hy  m  -\-  n. 

13.  a  —  b  by  a  —  b.  14.    r  +  »  by  r  —  8. 

15.  3  7W  4-  w  by  2  m  +  w.        16.    3  r  +  2  «  by  r  —  3  «. 

17.  5a:  +  4y  by  3a;  +  2i/.      18.    8m— 2n  by  2  m-h3ri. 

19.  3a;2H-2yby4a;-8«/2.      20.    3^24.4  ^3 by  2^2-4  w2 

21.  2  a:^  -f-  3  2  by  5  x^t/  —  4:Z. 

22.  5 3^t/^-2z^  by  Sa^f-^z\ 

23.  6a:3_i  by  22^3+1. 

24.  5  a^^2  _  1  by  6  a:2^2  _  1^ 

25.  m2  —  7WW  +  7l2  by  w  +  71. 

26.  2x^-\-^xi/  +  4:2/^hy  x+1/, 

27.  0:2  H-  a:^  +  i/2  by  ic  +  y. 

28.  2^2  _|.  ^2-  +  a2  by  2  a:  +  3  a. 

29.  4r2  +  6r«  +  9«2  by  2r4-3«. 

30.  3ic3^  2a:-l  by  4a;2+2. 

31.  4  a*  -  3  a2  +  a  by  2  a3  _  3  a\ 

32.  mV  +  2  WW  +  1  by  ww  H-  1. 

33.  2:2^2  -f  4  2;y  +  4  22  by  a;y  4-  2  0. 

34.  2a^  +  Sx^-4x-Shy^a^-h2x. 

35.  2a:34.3a;2_^^4by  22^-82:  +  2. 

36.  2^2  ^  2:^  ^  ^2  by  2^2  _  2;^  _|_  y2^ 

37.  3  m2  -  2  m  +  1  by  m2  4-  3  m  +  2. 

38.  2^  +  2a;+2by2:2_2rr  +  2. 

39.  4  2^2  ^12^^  + 9  ^^2  by  2 2;+ 3^. 

40.  2^  +  a^^  +  2:?/2  ^  y3  by  2;  —  y. 

41.  82:34. 122^^4. 182;2/2  +  27^by  22:-32/. 

42.  6  TwV  —  2  7w2w2  4-  4  7WW  —  1  by  3  mV  —  2  mn  —  3. 


64  ELEMENTARY  ALGEBRA 

Expand : 

43.  (4a2+ 2a  +  l)(4a2_2a-|-l). 

44.  (9^2- 6mn-f-4w2)(9w2+6wn  +  2w2). 

45.  (a^-x-l}(2^-2a^-x-{-l). 

46.  (a;-l)(a;-2)(2;-3).  47.    (x^-x  +  iy.  ' 
48.    (3^-^2x^-\-Sx-j-iy.  49.    (2m4-47i  +  J)2. 

50.  (2r-«)(r  +  2«)(2r  +  8«)- 

51.  (m  +  w  +  ^)^.                       52.  (a  —  6  —  c)8. 
53.  (a;»4-2)(2:«  +  3).                 54.  (a;»  -  4)2. 
55.  (2 a +36)3.                         .56.  (3  m- 2/1)8. 

57.    (a>4-4)(ir2n_4).  53.    (2  a  +  3  6  -  c-H  4  (?)2. 

59.  {T^-^-T^y-^-x^y^-^-xy^^-y^^ix-y), 

60.  (3a;-2y  +  42-l)2. 

61.  What  is  the  area  of  a  rectangle  that  is  2  a  +  8  units 
long  and  3  a  -f  1  units  wide  ? 

62.  By  how  much  is  the  area  of  a  rectangle  of  base  h 
and  altitude  a  changed  by  increasing  the  base  by  2  and 
decreasing  the  altitude  by  1  ? 

Division 

62.   The  quotient  of  two  powers  of  the  same  base. 

Since  a2  x  a^  =  a^,  it  follows  from  the  definition  of  divi- 
sion that 

a^  -J-  a2  _  ^3^  Qp  ^6-2 

In  general,  when  w  and  w  are  positive  integers  and  n  is 

less  than  w, 

0^  -5-  a"  =  a"~**. 

The  exponent  of  the  quotient  of  two  power%  of  the  same 
base  is  equal  to  the  exponerit  of  that  base  in  the  dividend 
minus  its  exponent  in  the  divisor. 


FUNDAMENTAL  PROCESSES  65 

EXERCISE  27 

Name  the  quotients  in  the  following  examples : 

1.  c?  -h  a.  2.    a^  -J-  a^.  3.   y^  -j- 1/^. 

4.  a^^2^.  5.     R^-i-E.  6.    m^-^m\ 

7.  (_^)^(4.a:4).  8.    (a^)^(i-a^). 

9.  (_a;i2)^(_^2).  10.    58-^52. 

11.  45-43.  12.    (25) -(-23). 

13.  (ia-^by-T-(a-^by.  14.    (m  +  ti)^— (m  4- ri)*- 

15.  (x-ht/y-^Cx-ht/y.  16.    (rH-l)6-(r+l)8. 

17.  (l  +  a)7-(l  +  a)2.  18.    (jt?  +  ^)io-(^  4-^)6. 

19.  (2  2:+«/)i<^^(2  2;4-?/)^.      20.    (a  +  ^-0^-^(«  +  ^-<?)- 

21.  (2a  +  a;  — i/)7-;-(2a  +  2:— «/)2^ 

In  examples  22-30,  the  literal  exponents  are  assumed 
to  be  integers  and  the  exponent  in  the  divisor  in  each  case 
is  assumed  to  be  less  than  the  exponent  in  the  dividend. 

22.  x° -i- x^.  23.    x'^-^x. 
24.    a^^  H-  a"".  25.    ty^^  -r-  t/"". 
26.     —  a*"  -r-  a^.  27.    a**  —  (—  a»). 
28.    (-w°)-5-(- w*).  29.    r3^^(-r^). 
30.    a^+1  -^  a^-i. 

63.  Meaning  of  zero  exponent.  The  definition  of  expo- 
nent given  in  section  15,  page  11,  does  not  apply  to  zero 
used  as  an  exponent.  When  a  is  any  number  other  than 
zero  we  shall  assume  that 

Remark.  It  has  been  shown  in  section  62  that  a""  -^  a^  =  a*"-", 
when  m  and  n  are  positive  integers  and  n  is  less  than  w.  If  n  is 
equal  to  m,  this  equation  becomes  a^  -^  a"*  =  a"*"*"  =  a^ ;  but  a"*  -f-  a"* 
=  1.     Hence,  if  a""  -^  a"*  =  a"*"**,  in  which  a  is  difPerent  from  zero,  is 


66  ELEMENTARY  ALGEBRA 

an  identity  when  n  =  m,  it  is  necessary  to  make  the  assumption  given 
in  section  63 ;  namely,  that  a9  =  1. 

EXERCISE  28 

1.  Explain  why  aP  =  h^. 

2.  Explain  why  20=50  =  ( J)0. 

Express  in  the  simplest  form : 

3.  6^aO;  c^a%^;  a%^c^ -h  c\ 

4.  a'^b^-i-a^;  a%^-^h^(P;  m^n^^n^r^. 

5.  a%^-^(P;  ^ah^^a%\  a^m^bHm'^. 

64.  Division  of  monomials.  Since  division  is  the  inverse 
of  multiplication,  and  since 

2a52^x  3a2J3^  =  6a356^, 
it  follows  that  6  a%^(^  -5-  2  ah^c  =  3  ^253^ 

and  6  a^JV  ^  3  ^253^  ^  2  a52c. 

Also,  since  —  8  o^yz  x  2  a:y  =  — 16  a^y^z^ 

it  follows  that     —  16  o^yH  -. —  8  x^yz  =  2xy 
and  —  16  2:^^225  -i-2xy  =  —  ^ x^yz. 

From  these  examples  we  infer  the  following  rule  for 
finding  the  quotient  of  two  monomials : 

Rule.  Divide  the  numerical  coefficient  of  the  dividend  hy 
that  of  the  divisor  (^observing  the  laws  of  signs  for  division) 
and  write  after  the  quotient  each  letter  of  the  dividend^  giving 
it  an  exponent  equal  to  its  exponent  in  the  dividend  minus 
its  exponent  in  the  divisor. 

Note  1.  If  there  occurs  in  the  dividend  any  letter  which  is  not 
found  in  the  divisor,  it  may  be  understood  to  occur  in  the  divisor 
with  an  exponent  0.  If  the  same  "power  of  a  letter  occurs  in  both 
dividend  and  divisor,  the  difference  of  its  exponents  being  0,  the 
letter  is  omitted  from  the  quotient ;  for,  a^  =  1. 

Note  2,     The  division    of    one   monomial   by  another   may   be 


FUNDAMENTAL  PROCESSES  67 

expressed  in  various  ways.     Thus,  10  x^y^  -=-  5  xy^  =  2  x^y  may  also  be 
expressed : 

l?^£^  =  2:.3y   or    5:ry^)10:rV, 
5x2/2  ^  2x^y         ■ 

Note  3.  When  the  numerical  coefficient  in  the  dividend  is  not 
exactly  divisible  by  that  in  the  divisor,  the  numerical  coefficient  in 
the  quotient  is  an  arithmetical  fraction  and  should  be  simplified  as 
in  arithmetic. 

Thus,  9  a%^  ^  6  a62  =  I  ah,  or  |  ad. 


EXERCISE  29 

(Solve  as  many  as 

possible  at  sight.) 

Divide : 

1.    ^7^hj2x. 

2.    -  6  a6  by  3  a2. 

3.    10  56  by  -2  63. 

4.    12a352by3a. 

5.    -  15  a%^  by  -  3  ah\ 

6.    Ux'h^yhj4.h^y. 

7.    -  21  a%^c  by  7  a%^. 

8.    —  10  2:^225  by  —  2  xyz. 

9.    lIx^yH^hy  -2xy^z. 

10.    81a35cby9A. 

11.    '^'^x^y^z^hyl^xyz. 

12.    96  2^3^225  by  -^x^yH. 

13.  8(a  +  6)3  by  2(a  -f  6). 

14.  -12(a:H-2^)*by  -(a:H-y). 

15.  -  1 2  a553(^  +  ^)2  by  -  3  a^js^^  4.  ^). 

16.  15  a:S«/225(^  4.  ^-)6  by    _  5  a:Sy2(^  _|_  2;)4. 

17.    I  Trigs  by  7ri22,  18.    |  tt^s  by  i  B. 

In  examples  19-26  the  literal  exponents  are  assumed  to 
be  integers,  and  the  exponent  in  the  divisor  in  each  case 
is  assumed  to  be  less  than  that  in  the  dividend. 

19.    6  a*  by  2  a?i.  20.    9  a^  by  —  3  a. 

21.    -  12  X"'  by  4  X».  22.    - 15  X^  by  15  X"». 

23.    12  a^+i  by  —  4  a.  24.    16  or  by  4  a"»-2. 

25.    -  18  a"»+i  by  9  a"»-i.  26.    20  a^+i  by  -  5  a*. 


68  ELEMENTARY  ALGEBRA 

65.  Division  of  a  polynomial  by  a  monomial.  Since 
2  a(3  b  —  4:c-\-6  d)  =6  ab  —  Sac  + 10  ad^  and  since  divi- 
sion is  the  inverse  of  multiplication,  it  follows  that 

(6  a6  -  8  ac  +  10  a<f)^  2  a  =  3  6  -  4  C+  5  <? 

_^ab      8  gg      10  at? 
""2a       2a         2a   ' 

From  the  foregoing  illustration  we  infer  the  following 
rule  : 

Rule.  To  divide  a  polynomial  by  a  monomial,  divide  each 
term,  of  the  polynomial  by  the  monomial  and  add  the  quotients. 

Note.     (Qab  —  8ac  + 10  ad)  -r-  2  a  may  be  written 

Qab-Sac  +  lOad 
2a 
or  2  a)Q  ah  -  8  ac  -f  10  ad. 

EXERCISE  30 

(Solve  as  many  as  possible  at  sight.) 
Divide : 

1.    3  a2  -  6  a  by  3.  2.    a^  +  ab  by  a. 

3.    6a^-9xhySx.  4.    2a2-2a6by2a. 

5.    6  x^y  —  9  xy^  by  3  xy.  6.    x^yh'^  —  xyz  by  xyz. 

7.  4a2-652_8c2by  2. 

8.  6  m^  —  8  mn  4-  4  mn^  by  2  m. 

9.  3  a^bx  —  2  ab^x  —  3  a^ex  by  aa;. 

10.  15  x^y  —  20  a;^2  _  5  g.^  by  5  a;y. 

11.  a^  —  2  a  by  —  a. 

12.  2  aSJ  -  3  a52  +  2  aft  by  -  aft. 

13.  3  a^yz  —  5  2:^%^  _j_  2  xyz^  by  ~  xyz. 


FUNDAMENTAL  PROCESSES  69 

Perform  the  indicated  divisions  : 

,^     irRL^irm  „     27r^+7rig3 

14- -•  -I-***     :;:: * 

7  rn^n  —  14  n^  +  21  mm 
16.     = • 

7  mn 

ab(m  +  ny  +  ah^jm  +  ^) 
a6 


18. 


19. 


20. 


{m-\-Yi) 

—  18  mhfir  —  12  m?tV  +  6  mnr 
—  6mnr 

.5r 


66.  Division  of  a  polynomial  by  a  polynomial.  Since 
division  is  the  inverse  operation  of  multiplication,  it 
follows  that  the  product  of  two  polynomials  divided  by 
one  of  them  is  equal  to  the  other.  Some  of  the  steps 
involved  in  dividing  one  polynomial  by  another  are 
suggested  by  a  careful  inspection  of  the  following  multi- 
plication : 

2  a;2  -  3  a:  +  4 
x^-     X  -  1 


2  a:*  -  3  a:8  +  4  a;2 

-2a:8  +  3a:2-4a: 

-2a:2  +  3a;-4 
2x4-5x8  +  5x2-     a: -4 

Let  it  be  required  to  find  a^  —  x  —  1^  given  2  a^  —  3  a;  +  4 
and  2x:^  —  5a^-{-5a^  —  X  —  4;  that  is,  to  find  the  quotient 
oi2a^-5a^-^5x^-x-4:  divided  by  2a^^  -  3  a:  +  4.  The 
quotient,  which  is  the  other  factor,  a^  --  ir  —  1,  may  be 
found  as  follows : 


70  ELEMENTARY  ALGEBRA 

X^-  X  -1 


1.  2x*-h2x^  =  x^  2x^-Sx-h4:)2x*-5x»  +  5x^-x-i 

2.  Subtract  (2  a;2  -  3  ar  +  4)  X  a:2  =  2x*-dx^  +  ^x^ 


5.  -2x»-h2x^  =  -x  -2x»+x^-x-4: 
4.    Subtract  (2  ^2  -  3  a;  +  4)  x  (-  x)  =  -2x»  +  dx^-ix 

6.  -2x2-2x2  =  -!  -2x2  +  3x-4 
6.    Subtract  (2x2- 3x  + 4)  (-1)=  -2x2  +  3x-4 

0 

Explanation.     The  explanation  of  the  process  may  be  given  thus : 

The  terms  of  both  dividend  and  divisor  are  arranged  according  to 
the  descending  powers  of  x. 

The  term  of  highest  power  in  the  dividend  being  the  product  of 
the  term  of  highest  power  in  the  quotient  and  that  of  the  highest 
power  in  the  divisor,  the  term  of  highest  power  in  the  quotient  is 
obtained  by  dividing  the  term  of  highest  power  in  the  dividend,  2  x*, 
by  the  term  of  highest  power  in  the  divisor,  2  x2.  This  gives  x2,  the 
term  of  highest  power  in  the  quotient. 

The  dividend  is  formed  by  multiplying  the  divisor,  2  x2  —  3  x  +  4,  by 
each  term  of  the  quotient  and  adding  the  resulting  products.  Hence, 
reversing  the  process,  2  x2  —  3  x  +  4  is  multiplied  by  x2,  and  the  result, 
2  X*  —  3  X*  +  4  x2,  subtracted  from  the  dividend,  2  x*  —  5  x*  +  5  x* 
—  X  —  4. 

The  remainder,  —  2  x*  +  x2  —  x  —  4,  must  be  the  product  of  the 
divisor  and  the  part  of  the  quotient  to  be  found.  Hence,  the  term 
of  highest  power  in  the  remainder  must  be  the  product  of  the  term 
of  highest  power  in  the  divisor  and  the  term  of  the  quotient  of  next 
higher  power  to  that  already  found.  Therefore,  the  term  of  next 
higher  power  in  the  quotient  is  obtained  by  dividing  the  term 
of  highest  power  in  the  remainder,  —  2  x^,  by  2  x2.  This  gives  —  x, 
the  term  of  next  higher  power  in  the  quotient. 

Multiply  the  entire  divisor  by  the  second  term  of  the  quotient,  —  x, 
and  subtract  the  product,  —  2  x*  +  3  x2  —  4  x,  from  the  remainder. 
This  leaves  —  2  x2  +  3  x  —  4,  the  second  remainder. 

The  third  term  of  the  quotient  (—  1)  is  obtained  from  the  second 
remainder  just  as  the  second  term  is  obtained  from  the  first  re- 
mainder. The  entire  divisor  is  then  multiplied  by  ( —  1)  and  the 
product,    —  2  x2  +  3  X  —  4,   subtracted  from   the   second  remainder. 

The  final  remainder  being  zero,  the  division  is  said  to  be  exact. 

The  quotient  is  x2  —  x  —  1. 


FUNDAMENTAL  PROCESSES  71 

From  the  foregoing  explanation  we  may  derive  the 
following  rule  for  dividing  one  polynomial  by  another : 

Rule.  1 .  Arrange  the  dividend  and  the  divisor  according 
to  the  descending  or  ascending  powers  of  the  same  letter. 

2.  Divide  the  first  term  of  the  dividend  by  the  first  term 
of  the  divisor  ;  this  gives  the  first  term  of  the  quotient. 

3.  Multiply  the  entire  divisor  by  this  first  term  of  the 
quotient^  and  subtract  the  result  from  the  dividend  ;  this  gives 
the  first  remainder. 

4.  Divide  the  first  term  of  this  remainder  by  the  first  term 
of  the  divisor  ;  this  gives  the  second  term  of  the  quotient, 

5.  Multiply  the  entire  divisor  by  this  second  term  of  the 
quotient.,  and  subtract  the  result  from  this  remainder ;  this 
gives  the  second  remainder ;  and  so  on. 

Remark.  Divisor,  dividend,  and  successive  remainders  should 
always  be  arranged  in  the  same  order  of  powers  of  the  same  letter. 

ILLUSTRATIVE  EXAMPLES 

1.    Divide  2a3_4«_5a2  +  3by2a-l. 

Arrange  the  terms  according  to  the  descending  powers 

of  a. 

a2  _  2  a  -  3 
2a-  1)2 «»  -  5 «2  _  4  a  +  3 
2  a8  -     a^ 

-4:a^-]-2a 


-6a  +  3 

-6a  +  3 

0 

Check.     Let  a  =  1. 

Divisor^  2  «  —  1  =  2  —  1  =  1. 

Dividend,  2a3-5a2-4a  +  3  =  2-5-4  +  3=-4. 

Dividend  -^  Divisor  =(—  4)-=-!=  —  4. 

Quotienty  a^-2a-3  =  l-2-3=-L 


72  ELEMENTARY  ALGEBRA 

Remark  1.  No  number  should  be  used  in  checking  which  when 
substituted  makes  the  divisor  zero. 

Remark  2.  Observe  in  the  solution  of  example  1  that  only  the 
part  of  the  remainder  about  to  be  used  is  brought  down  at  each  stage. 

2.    Divide  a^  -\- y^  hj  x -\-  y . 

x^  —  x^y  -f  xy^  —  y^ 


X  +  y)x^ 

+  t 

x^  +  x^y 

-x^y 

-^y 

-xY 

x^ 

xY^xf 

-xy^^y^ 

-  ^f  -  f 

2v* 

Remark.  The  quotient  is  3^  —  x^y  +  xy^  —  y^ ;  the  remainder  is 
2  y\  The  complete  quotient  may  be  expressed  in  the  same  way  as  in 
arithmetic.     In  this  instance  the  complete  quotient  may  be  written, 

2  u* 
X  -^y 

Check.  Let  a:  =  1,  and  y  =  1. 

Dividend  f  a;*  +  2/*=l  +  l=2. 

Divisor,  x-\-y  =  l-\-l=2. 

Dividend  -i-  Divisor  =2-^2  =  1. 

Quotient,  a;»  -  x'^y  +  xy^  -  y^  +  -^^  =1~1  +  1-1+| 

x-\-  y  2 

=  1. 

Another  Check.     As  in  arithmetic,  if  the  division  is  correct, 
Divisor  x  Quotient  +  Remainder  =  Dividend. 

Hence,  if  we  evaluate  both  members  of  the  equation  for  the  same 
numerical  values  of  x  and  i/,  and  the  results  do  not  agree,  we  shall 
know  that  there  is  an  error. 


FUNDAMENTAL  PROCESSES  73 

3.    Divide  abx^  -\-  (a^  -f-  h'^)x  H-  ah  by  ax  +  h. 
Solution 

hx   -\-  a 
ax  +  h)ahx'^  +  a^x  +  5%  +  ah 

abx^ +  h'^x 

a^x  +  db 

a^x  +  db 


Check 

Let  X  =  1,  a  =  2,  and  5  =  3. 

Then,         a&a:2  -\-a^x  +  6%  +  a6  =  6  +  4  +  9  +  6  =  25. 
ax  +  6  =  5,  and  6x  +  a  =  5. 
^  =  5. 

EXERCISE  31 

Divide  : 
1.    x'^-^^x-\-2hy  x-\-l,  2.    a:2_^5^^g  by  ^^3^ 

3.  x^-\-lx-\-12hyx  +  ^.  4.  a2^3^_,_2  by  aH-2. 
5.  2;2- 9 a: 4- 20  by  2: -4.  6.  ^2_  7^  ^  6  by  ^  _  1^ 
7.    a;2  —  a:  —  20  by  a:  —  5.  8.    m^  +  2  w  — 15  by  ?7i  +  5. 

9.    r2  +  6r- 7  byr+ 7.         10.    t;2  -  4t;  -  21  by  v- 7. 

11.    x^  +  x-^Ohy  x  +  Q.         12.    ^2  _  ^  _  2  by  ^  +  1. 

13.    2:4-62:24.6  by  0^2  _  2.       14.    2:4  +  2:2  _  12  by  2:24.4. 

15.    9  4-10^4-^2  by  1  4- w.   16.    4  -  3  A:  -  A:2  by  4  4- )fc. 

17.    12-8r4-/^by  2-r.       18.    42:2  -  1  by  22:4- 1. 

19.    9  w2  _  4  by  3  w  4-  2.  20.    rri^  —  n^  hy  m -\- n. 

21.    2:2  —  2  2:y  4- y^  by  2:  —  y.     22.    m2  4.  2 77m  4- ^2  by  w  +  w. 

23.  ^2  4.  2^^-1-1  by^z  +  l. 

24.  2a2+ii«5_f.l252by  2a4-36. 

25.  6  ^2  —  WW  —  12  n2  by  3  w  4-  4  w. 

26.  10r24-16r»-882by  5r-2«. 


74  ELEMENTARY  ALGEBRA 

27.  36  ?/2  _j.  24  a«/  —  5  a2  by  6  3^  -h  5  a. 

28.  2a:2^2_^a:i/-6by  2a;y-3. 

29.  m^— 2m/i  — 15^2  by  m  — 5?i. 

30.  6  a:* -11  2:2 -35  by  82:24.5. 

31.  a:^  _j.  3  2;2  _|_  4  a;  _|.  4  by  a:2  ^  a;  +  2. 

32.  4a:3^i0a^+4a:-2  by  2rr2+3a:-l. 

33.  10^3_37^2^13^_21  by  5^2_^^3^ 

34.  2«3  +  5a2_f.4^_,_l  by  2a  +  l. 

35.  8m3+7m2-10  +  9mby  3w-2. 

36.  8y3_l9y_lo^2_i5by  2^-5. 

37.  a;^  —  ?/3  by  a;  —  ?/. 

38.  a^*"  +  5^  by  a"*  +  h^. 

39.  a:*  4-  a:2y2  _j_  ^^  by  2:2  +  a:^  -j-  y%^ 

40.  a3_^  1  by  a4-l. 

41.  8  w3  4.  27  by  2  w  4-  3. 

42.  8«/3_64  by  23/ -4. 

43.  m^  +  8  7712/1  4-  3  mr^  4-  ti^  by  w  4-  w. 

44.  m*  +  w2  4-  1  by  w2  —  w  4-  1  • 

45.  20  r^  -  36  r^s  4-  59  tH^  -  57  r^s^  4.  29  r*^  -  15  «6  by 
10r2-3r«4-5«2. 

46.  7!^  —  y^hy  X  —  y.  47.    w^  H-  ^i^  by  ?w  +  w. 
48.    w2  4-w^byw  — w.  49.    w*4-lbyw4-l. 

50.  2:8_|_  5^+ 102,^10  by  2:4-2. 

51.  6  w^  —  mhi  —  14  mw2  4-  3  w^  by  3  m2  4-  4  mn  —  n^. 

52.  2  a^  4-  8  a*  -  12  a3  4-  15  ^2  _  11  a  4-  3  by  -  8  a  4- 1 
4-  2  a2. 

53.  3/-16/4-1  by  3jp2-jt?4-l. 

6  1 

54.  Find  the  quotient  of  ^  ~     ;    check  the  result  by 
letting  a  =  2. 


FUNDAMENTAL  PROCESSES  75 

a^  —  1 

55.  Find  the  quotient  of  — ;    check  the  result  by 

letting  a  =  —  2.  ~ 

56.  Find  the  quotient  of  -— ;  check  the  result  by 

letting  a  =  1. 

57.  Find  the  quotient  of  — -4-  ;    check  the  result  by 
letting  a  =  1.  ^ 

58.  What  is  the   remainder  when  a*  —  a^  +  2  a  +  1  is 
divided  by  a2_2a  + 3? 

59.  For  what  value  oi  m  is  a*  —  a^  —  8  a^-\-  ma  —  3  ex- 
actly divisible  by  a^  -f  2  a  —  3  ? 

60.  For  what  value  of  m  is  a^  -\-  a^ -\- 2  a^ -\-  a^ -\- ma  —  2 
exactly  divisible  by  a^  —  a^  +  2  a  —  1  ? 

61.  What    are    the    values    of    Q    (quotient)    and    E 
(remainder)  in  the  following  expression: 

a  +  1  ^^a+1' 

62.  Show  that  -^  =  l+aH-a2  4.^8  +  ^4^     ^ 


a  1—  a 

63.    If   a  =  — ,  what  error  is  made   in   assuming   that 


1-a 

EXERCISE  32.  — GENERAL  REVIEW 

1.  Find  the  algebraic  sum  of  a:,  4  y,  and  —  32. 

2.  Find  the  algebraic  sum  of  3(a;— ^),    —  4(a;— y), 
2(x  -  y\  and  -  b(x  -  y). 

3.  Simplify    ^x  —  2a  +  ^h  —  2c—^a^2x—h-\-c 
-  X  -2h  +  ^a-{-^c. 

4.  Simplify   ah  +  2  a%  -  ab^ -h  2  ab^  ^  S  a%  -  4  a^I^ 
+  5  aJ-  3  ab^  •{.  a%  -  6  ab  +  aW  +  5  ab^  -f  3  ^252. 


76  ELEMENTARY  ALGEBRA 

5.  Add  -  3(a  -  b),  4(<i  +  b),  -  5(aH-5),  and  6(a-6). 

6.  Add  ^  x'i/  +  ^  x^i/-\-l  xf-  .1  3^1/^  ^xy^-lxy-^x^y 
+  .7  x^y\  \xy  —  .3  a^y^  and  x^y  —  \  xy^. 

7.  What   must   be   added    to    a^—2ab-{-  5^    to    give 

8.  What  must  be  added  to  x^  -{-  S  xy  —  2  y^ -\-  4:  to  give 
-Sx^  +  xy-y^-2? 

9.  What    must    be    added    to  x^-^  2xy  +  y^   to  give 
x^  -  2xy  +  y^? 

10.  r2-2r-7-(-2r2+r-3)=? 

11.  Simplify  l-{_l-[-l_(-l)_l]-l}  +  l. 

12.  Inclose  the  last  three  terms  oix  —  y  —  z  +  1  within 
parentheses  preceded  by  the  sign  — . 

13.  Name  at  sight  the  products  : 

ab  '  ab;  a%  •  ab'^ ;  a"5"  •  ab  ;  a"5"  •  a'^b^  ;  2  ax""  ■  3  a^. 

14.  Expand  (a+  l)(a  +  l)(a  -  l)(a—  1). 

15.  Multiply  2  a2  _,.  6  a6  +  3  52  by  2  a2  -  6  a6  H-  3  b\ 

16.  Simplify(2a:  +  3«/)(2a;+3t/)-(2a:-3y)(2  2:-3y). 

17.  Multiply  (2m-\-lfhy  (2m-  1)2. 

18.  Evaluate  2  irRCH  +  i2)  in  terms  of  ir  when  i2  =  2 
and  ir=  3. 

19.  Evaluate  irR^L  -f  i2)  in  terms  of   ir  when  i2  =  3 
and  X  =  5. 

20.  Evaluate  ^  irH^B^  -\-  Rr-\-  r2)  in  terms  of  tt  when 
ir=:6,  i2  =  4,  r  =  l. 

21.  Evaluate  J  irH^E^  H-  i2r  +  r2)  in  terms  of  tt  when 
5^=9,  i2  =  3,  r=0. 

22.  Evaluate  (mn  —  r8)(n8  —  mr)(nr  —  ms)  when  7?i  =  2, 
71  =  —  1,  r  =  0,  and  «  =  1. 


FUNDAMENTAL  PROCESSES  77 

23.  Name  the  quotients  in  the  following : 

a^^a?]  w>^w>\  a^'^a'"',  26-5-23. 

24.  Name  the  quotients  in  the  following  : 

a%^^  aV)  ',a%-^ah\   -  6  a^J*-^  2  ^352 .  _  (3  ^^)3^  _  (3  ^5)2. 

25.  Name  the  quotients  in  the  following  : 

a**6"  -r-  ah  ;  x^^^y^^'^  -?-  x^y^ ;  ic^"^  -^  a;"y. 

26.  Divide  irW'  by  1  R. 

21,    Divide  2  irRH-^  2  7ri22  by  27ri2. 

28.  Divide  irRL  +  irR^  by  ttR. 

29.  Divide  a25  _  aja  -|-  2  aW  by  -  a6. 

30.  Divide  (a  -  6)2;  +  (a  -  h^x^-  (a  -  6)(a  +  6)2:8  by 
(a  —  h')x, 

31.  Divide  m^  —  n^  hy  m  —  n. 

32.  Divide  1  -|-  2  a;  by  1  -|-  2:,  carrying  the  quotient  to 
four  terms. 

33.  Divide  a^+b^  +  ^  aH  -\- ^  aW'  by  a  +  6. 

34.  Divide  a*  -^  5*  -  4  aft^  _  4  a%  -i-  6  a252  by  a  -  6. 

35.  Express  m— n— 'p-\-q  —  r— ar^h—x— y  in- tri- 
nomial terms  having  the  last  two  terms  of  each  trinomial 
inclosed  within  parentheses  preceded  by  the  sign  — . 

36.  Show  that  (m  —  n)(^p—  q)  =  {n  —  m)(^q  —  p), 
31.    Show  that  (a  -  6)2=  (h  -  ay. 

38.  Show  that  gi:^'=^'-^'. 

a  —  0        h  —  a 

39.  Multiply  W2:3  _  ^^  j^  ^x  —c  by  moi^  +  2:^  -|-  4  2:  +  d 
and  arrange  the  result  according  to  powers  of  x. 

40.  In  the  formula  V=  ^ttR^^  find  the  value  of  V 
when  7r=  3.1416  and  R  =  8.' 

41.  Divide  3  a^  +  a*  -  12  a^  -|-  5  a2  -  15  a  -  6  by  a^ 
4-2a2  +  3. 


78  ELEMENTARY  ALGEBRA 

42.  Divide  3  aa^  -  U  aV  _  7  ^z^  ^  5  ^i^  _  q  ^6^2  by 
6  xS  -  3  aa^  +  2  A. 

43.  Multiply  ^2  +(a  +  ft)a;  +  a^  by  2;  — a,  arranging  the 
result  according  to  powers  of  x. 

44.  Show  that  Cax  +  bi/y+(ibx-ayy=(a^-\-b^){2^-^i/^). 

45.  If  (x  +  a)(3  a^  -  2  5a;  +  3)  is  the  same  as  3  a^s  _  ^^ 
—  a;  +  3,  find  a  and  b. 

46.  The  expression  aa^i/^  has  the  value  32  when  a;  =  2 
and  t/  =  1  ;  find  its  value  when  a;  =  —  3  and  y  =  —  2. 

47.  Find  the  value  of  a^-S  a«+i  +  2(a  +  1)«  when 
a=2. 

48.  Find  the  value  of  2a;—  3^+z  when  a;  =  a  +  ft  —  c, 
?/=a— 5+2c,  and  z=Sa-\-2b  +  c. 

49.  From  the  formula  s  =  at  -\-  ^ft^^  find : 

(1)  the  value  of  «  when  a  =  12,  f  =  2,/=  32. 

(2)  the  value  of/  when  «  =  64,  a  =  64,  e  =  2. 

50.  To  -  -h  -  add  ^  —  -and  take  -a  —  ^b  from  the  sum. 

51.  Simplify  3(a;-y)-2(y-2)-4(t-a;)-7(a;+y+0- 

52.  Find  the  value  of  n  which  satisfies  the  equation, 
7(58  -  w)  =  3(n  -  14)  -  14(n  -  25). 

53.  Subtract  2a  +  36  —  4c  from  5a  —Sb  +  2c and  add 
the  remainder  to  —  3  a. 

54.  Simplify  : 
4}3(|a-f6)+2(f6-i.)+6(fc-|a){. 

55.  Find  the  value  of  [p^  +  (a  +  l)jt>  +  a]  -*-  (/>  +  a). 

56.  A  man  has  to  walk  to  a  place  m  miles  away.  How 
far  will  he  be  from  his  destination  in  a  given  number  of 
hours  (f)  if  his  rate  of  walking  is  r  miles  an  hour  ? 

57.  Evaluate  a""^,  also  a"  —  1,  when  a  =  3  and  n  sa  5. 


FUNDAMENTAL  PROCESSES  79 

58.  Find  the  product  oi  x  —  2y,  x-\-2y^  and  x  —  ^y. 

59.  In  the  year  1900  a  man  on  his  birthday  found  that 
the  number  of  months  he  had  lived  was  half  of  the  date 
of  the  year  of  his  birth  ;  how  old  was  he  ? 

60.  If  (a  +  l)x -{-l=x-\-2a,  express  x  in  terms  of  a. 

61.  A  man  has  four  sons  whose  combined  ages  are  equal 
to  his  own.  In  20  years  their  combined  ages  will  be 
double  the  age  of  their  father ;  what  is  his  present  age  ? 

62.  Show  that  S  a^ -\- b^ -\-  (^  —  6  abc  is  exactly  divisible 
by  2  a  H-  6  +  c. 

63.  Show  that 

64.  The  volume  of  a  circular  cone  is  given  by  the 
expression  ^  ttt^A  where  h  and  r  represent,  respectively,  the 
measures  of  the  height  and  the  radius  of  the  base  in 
terms  of  the  same  unit.  Find  to  two  decimal  places 
the  volume  of  a  cone  5  ft.  high  and  12  in.  in  diameter. 
(Take  w  =  3.14.) 

65.  Find  the  value  of  Z,  when 

a^-a-l  =  (^a-\-  2)(a -  3). 

66.  Find  the  value  of  m,  when 

a^  -  ma  —  35  =  (a  +  5)(a  -  7). 

67.  Find  the  value  of  w,  when 

^a2  +  3  a  -  6  =  3(3  a  +  2)(2  a  -  1). 

68.  First  indicate  and  then  perform  the  following 
series  of  operations  :  To  the  product  of  (a  +  2  5)  and 
(2  a  —  6)  add  the  product  of  (a  —  2  b)  and  (2  a  -f  6),  and 
divide  the  sum  by  4. 

69.  If  a  gallon  of  diluted  milk  contains  p  pints  of 
water,  how  much  pure  milk  is  there  in  n  gallons  of 
the  mixture  ? 


80  ELEMENTARY  ALGEBRA 

70.  Representing  the  sum  of  two  numbers  by  «,  their 
difference  by  d^  the  larger  number  by  iV,  and  the  smaller 
by  n,  express  by  formulae  the  following  statements : 
"Half  the  sum  of  any  two  numbers  plus  half  their 
difference  is  equal  to  the  greater  number,  and  half  their 
sum  less  half  their  difference  is  equal  to  the  smaller 
number." 

71.  If  n  denotes  any  positive  integer,  what  kind  of 
integers  are  completely  represented  by2w?     By  2/1  —  1? 

72.  If  eggs  are  sold  according  to  quality  for  50  cents, 
42  cents,  and  38  cents  per  dozen,  write  a  formula  for  the 
total  cost  in  dollars  of  m  dozen  of  the  highest  grade  eggs, 
n  dozen  of  the  medium  grade,  and  p  dozen  of  the  lowest 
grade. 

73.  What  number  is  represented  hy  St^-\-2t-\- 1  when 
t  denotes  the  number  10  ?  Can  any  integer  of  three 
digits  be  expressed  in  the  form  at^  -\-bt  +  c  (read  from 
left  to  right)  where  a,  5,  and  e  denote  the  digits  used  to 
express  the  number  and  t  denotes  the  number  10  ? 

74.  Using  the  notation  of  problem  73,  write  the  six 
numbers  which  can  be  represented  by  using  the  digits  a, 
5,  and  c. 

75.  a  and  b  are  two  digits  of  which  a  is  the  larger. 
Find  a  formula  for  the  difference  between  the  two  num- 
bers which  can  be  represented  by  them. 

76.  If  a  represents  the  sum  of  the  ages  of  n  persons  b 
years  ago,  what  expression  represents  the  sum  of  their 
present  ages  ? 

77.  Divide  2  a;"»  +  3  rr'^+i  +  14  x^+^  -|-  32  x"'+^  -  98  ic«+4 
-|-39a:'«+5by  2  +  E> x - '6 x^. 


CHAPTER  III 
SIMPLE   EQUATIONS 

67.  Identity.  If  the  two  members  of  an  equation  are 
such  that  the  one  can  be  transformed  into  the  other,  the 
equation  is  an  identical  equation  and  is  called  an  identity. 

Thus,  3a  +  4a  =  7aisan  identity. 

Note.  Since  the  two  sides  of  an  identity  represent  ways  of  ex- 
pressing the  same  number,  the  statement  that  they  are  equal  is 
always  true,  whatever  values  may  be  assigned  to  the  letters  involved. 

Thus,  ^x  —  2x  =  X  is  true  whatever  value  be  given  to  x. 

68.  Conditional  equation.  An  equation  which  is  true 
only  when  the  letter  or  letters  involved  are  restricted  to 
certain  values  or  sets  of  values  is  called  a  conditional 
equation. 

Thus,  the  equation  x  —  1  =  0  is  true  only  when  x  has  the  value  1. 
The  equation  x  +  y  —  2  is  true  for  many  sets  of  values  of  x  and  y, 
but  not  for  all  values  of  x  and  y. 

Remark.  When  in  algebra  the  word  equation  is  used,  a  condi- 
tional equation  is  usually  meant. 

69.  Notation.  In  algebra  unknown  numbers  are  usually 
represented  by  the  last  letters  of  the  alphabet,  as,  w,  v,  a?, 
y,  z ;  and  known  numbers  by  the  first  letters,  as,  a,  6,  c. 
By  this  convention  it  is  easy  to  distinguish  at  a  glance 
between  the  known  and  the  unknown  numbers  which 
occur  in  the  equation. 

81 


82  ELEMENTARY  ALGEBRA 

70.  Satisfying  an  equation.  Any  set  of  values  of  the 
letters  of  an  equation  which  reduces  the  equation  to  an 
identity  is  said  to  satisfy  the  equation. 

Thus,  X  =  2  satisfies  the  equation  3  x  =  6 ;  x  =  2  and  y  =  1  is  a 
set  of  values  of  x  and  y  which  satisfies  the  equation  2x  -\-dy  =  7. 

71.  Simple  equation  in  one  unknown  number.  Any 
equation  which  can  be  put  in  the  form  ax -{-h  =  0  is  called 
a  simple  equation  in  one  unknown  number. 

Thus,  5x-\-2  =  3x  +  4:  is  a  simple  equation ;  it  may  be  written 
2x-2  =  0     [§25], 

Remark.  The  simple  equations  considered  in  this  chapter  con- 
tain only  one  unknown  number.  Observe  that  no  higher  power  of 
the  unknown  than  the  first  occurs ;  and  that  the  unknown  number 
does  not  occur  in  the  denominator  of  any  fraction. 

Note.  A  simple  equation  is  called  an  equation  of  the  first  degree, 
or  a  linear  equation. 

72.  Solution  of  an  equation.  To  solve  an  equation  which 
contains  only  one  unknown  number  is  to  find  all  values 
of  the  unknown  which  satisfy  the  equation. 

73.  Root  of  an  equation.  A  root  of  an  equation  which 
contains  only  one  unknown  is  any  value  of  the  unknown 
which  satisfies  the  equation. 

Thus,  2  is  .a  root,  and  the  only  root,  of  the  simple  equation 
3a; -6  =  0. 

74.  Equivalent  equations.  Two  equations  in  one  un- 
known are  said  to  be  equivalent  when  they  are  satisfied  by 
the  same  value  or  values  of  the  unknowp ;  that  is,  when 
they  have  the  same  roots. 

Thus,  iix-\-S  =  2x-{-5,  and  j^  +  3  =  5  are  two  equivalent  equa- 
tions ;  each  one  has  the  single  root  2. 


SIMPLE  EQUATIONS  83 

Two  equations  are  therefore  equivalent  when: 

1.  Every  solution  of  the  first  equation  is  a  solution  of  the 
second. 

2.  Every  solution  of  the  second  equation  is  a  solution  of 

the  first. 

Thus,  the  two  equations  a;  -  1  =  0  and  a:^  -  x  =  0  are  not  equiva- 
lent; for  although  the  one  solution  of  the  first  equation,  namely, 
a;  =  1,  is  a  solution  of  the  second,  yet  the  second  equation  has  a  solu- 
tion, namely,  a:  =  0,  which  is  not  a  solution  of  the  first. 

Note.  A  simple  equation  is  solved  by  transforming  it  into  an 
equivalent  equation  which  shall  contain  the  unknown  number  alone 
in  one  member  and  its  value  in  the  other. 

Thus,  the  equation  3a:-f-3  =  2a:  +  5 

is  equivalent  to  x  -}-  3  =  5, 

which  is  equivalent  to  a:  =  2. 

75.  Transposition.  An  important  principle  follows  from 
assumptions  1  and  2,  section  25. 

Thus: 

Let  ax  —  b  =  c.  (1) 

Adding  b  to  each  member  of  equation  (1), 

ax-  b  +  h  =  c  +  b.  (2) 

Combining,  ax  =  c  -\-  b.  (3) 

Observe  that  equation  (3)  differs  from  equation  (1)  in  that  the 
term  containing  b  is  in  the  second  member  of  (3)  but  in  the  first 
member  of  (1),  and  that  the  signs  of  the  terms  containing  b  are 
different  in  the  two  equations. 

Again,  let  ax  -\-  b  =  c.  (1) 

Subtracting  b  from  each  member  of  equation  (1), 

ax  +  b  -  b  =  c  -  b.    ■  (2) 

Combining,  ax  =  c  —  b.  (3) 

Compare  equations  (1)  and  (3)  and  observe  as  before  that  the 
term  containing  b  is  in  the  second  member  of  (3)  but  in  the  first  mem- 
ber of  (1),  and  that  the  signs  of  the  terms  containing  b  are  different 
iu  the  two  equations. 


84  ELEMENTARY  ALGEBRA 

It  follows  from  the  foregoing  that : 
Any  term  may  he  transposed  from  one  member  of  an  equa- 
tion to  the  other,  provided  that  its  sign  is  changed. 

76.  Cancellation  of  terms  in  an  equation.  Cancellation 
of  terms  in  an  equation  may  be  illustrated  by  solving  the 
equation, 

x  +  b  =  c-\-b.  (1) 

Transposing,  x  =  c  -\-  b  —  b.  (2) 

Combining,  x  =  c.  (3) 

Comparing  (1)  with  (3)  we  may  infer  that : 
When  the  same  terms  preceded  by  like  signs  occur  in  both 
members  of  an  equation^  these  terms  may  be  omitted. 

77.  Change  of  signs  in  an  equation.  It  is  sometimes 
convenient  to  change  the  signs  of  all  the  terms  of  an 
equation.  This  may  be  done  by  multiplying  both  mem- 
bers by  -  1  [§  25]. 

For  example,  let  2  —  a:  =  —  5.  (1) 

Multiplying  both  members  of  equation  (1)  by  —  1, 

(-l)(2-:r)  =  (-l)(-5),  (2) 

or,  _  2  +  x  =  5.  (3) 

Observe  that  equation  (3)  is  equation  (1)  with  the  sign  of  each 
term  of  equation  (1)  changed. 

Remark.  It  is  evident  that  the  members  of  an  equation  may  be 
interchanged. 

ILLUSTRATIVE  EXAMPLES 

1.    Solve  the  equation  Sx  —  2  =  2x  +  5. 

Solution.  3  x  -  2  =  2  X  +  5.  (1) 

Transposing,  3  x  —  2  a:  =  5  +  2.  (2) 

Combining,  x  =  7.  (3) 

Check.  3  a:  -  2  =  2  a:  +  5.  (4) 

Substituting  7  ior  x,       3x7-2  =  2x7  +  5.  (5) 

Simplifying,  X9  =  19.  (6) 


SIMPLE  EQUATIONS  85- 

2.  Solve  the  equation  x -p-^  =  -(^x—  2). 

Solution.                        X  -  5^jt_?  =  hx-2).  (1) 

Multiplying  each  member  of  the  equation  by  the  least  common 
multiple  of  the  denominators, 

12.-lg(3^  +  2)^|(.-2).  (2) 

Performing  the  indicated  divisions, 

12ar  -  3(3  a:  +  2)=i(x  -  2).  (3) 
Performing  the  indicated  multiplications, 

12x-9x-Q=4x-S.  (4) 

Combining,                            3  a:  —  6  =  4  a;  —  8.  (5) 

Transposing,                             8  —  6  =  4  a:  —  3  a:.  (6) 

Combining,                                       2  =  a:.  (7) 

Interchanging  members,               a:  =  2.  (8) 

Check.                            x-^^^^  =  -(x-2).  (9) 

Substituting  2  fot  x, 

2_?Jil±-?  =  ?:(2-2).  (10) 

Simplifying,                           2-2  =  ^x0,  (11) 
or,                                                     0  =  0. 

3.  Solve  the  equation  2.5a;  —  3  =  .82:  +  2.1. 

Solution.                            2.5  a:  -  3  =  .8  a;  +  2.1.  (1) 
Expressing  the  decimals  in  (1)  as  common  fractions, 

i^-3  =  i^+fi.  (2) 

Transposing,                       |  a:  -  |  ar  =  -|1  +  3.  (3) 

Combining,                                 i^a:  =  ^.  (4) 
Multiplying  both  members  of  (4)  by  -J-^, 

a:  =  3.  (5) 

Check.                                 2.5  a:  -  3  =  .8  a:  +  2.1.  (6) 

Substituting  3  f  or  ar,     2.5  x  3  -  3  =  .8  x  3  +  2.1.  (7) 

Simplifying,                          7.5  -  3  =  2.4  +  2.1.  (8) 

Combining,                                  4.5  =  4.5.  (9) 


86  ELEMENTARY  ALGEBRA 

Remark.  Various  methods  of  procedure  may  be  resorted  to  in 
solving  example  3.  Thus,  each  term  of  equation  (1)  may  be  multi- 
plied by  10;  also,  .Sx  maybe  transposed  and  combined  with  2.5a: 
and  —  3  transposed  and  combined  with  2.1.  Again,  both  members 
of  equation  (2)  may  be  multiplied  by  10,  the  least  common  multiple 
of  the  denominators;  then  equation  (3)  would  be  replaced  by 
25a:-30  =  8x  +  21. 

EXERCISE  33 

Solve  the  following  simple  equations,  and  check  each 
solution  ; 

1.  13x-h7  =  5a;-4. 

2.  5u  +  2=2u-4. 

3.  13-6a  =  13a-6. 

4.  25c?-13  =  -6^  +  lll. 

5.  3m  +  2  =  llw-J^. 

6.  5jt?  +  12  =  17  -  5j9. 

7.  13r-ll  =  2r-ll. 

8.  15 -6f=  3^-12. 
Suggestion.     Divide  each  term  by  3. 

9.  -3^  +  17  =  125^-58. 

10.  13-lli/  =  133/  +  253. 

11.  2  +  3(a:-5)=5  +  4(a:-6). 

12.  3 -2(3^-4)  =5(2^^  +  3) -84. 

13.  3(jo  4-  2)  -  2(2jt?  -  3)  =  11  (3  -  7 jo)  +  72p  -  1. 

14.  11(1 -a:) +3(2 -a:) -5(3 -a:)  =11. 

15.  12«-5(3f-2)  =  3-2«. 

16.  3(2a;-3)=8-5(2a;-3). 

17.  5(z-3)+2(z-3)-4(2-3)=0. 

18.  2(3y-5)-7(23^  +  3)=5(3y-5)-8(2y  +  3). 
.19.  a;(a:+3)=r^  +  6. 


SIMPLE  EQUATIONS  87 

20.  (a:-hl)(a;H-3)  =  ar2-82:+27. 

21.  (a;4-3)(22;-5)=2a:(2;-2). 

22.  (^^-Hl)(y4-2)-(^  +  3)(y4■4)4-30  =  0. 

24.    3a!-Ka^+2)=8. 

2  5 

26.  i (3a; -1)=  11(2^ -7). 

27.  £l^-£ll2      £-_5^Q 

10  15  20 

28.  7-^^-(7x  +  l)  =  0. 

55  ^       77 

31.  .3a; +  4=. 9a: -2. 

32.  .8a;-l  =  .la;  +  2.5. 

33.  1.5a;- .5  =  . 7a; +.6. 

34.  .9a; -2.1  =  3.9 -.la;. 

78.  Solution  of  problems.  In  solving  a  particular  prob- 
lem which  leads  to  a  simple  equation  in  one  unknown 
number  it  is  necessary  to : 

1.  Restate  the  problem  in  algebraic  language  in  the  form 
of  an  equation. 

2.  Solve  the  resulting  equation  for  the  unknown  number, 

3.  Verify  the  solution. 


88  .  ELEMENTARY  ALGEBRA 

Remark.  Many  different  kinds  of  problems  occur  which  lead  to 
simple  equations  each  in  one  unknown  number,  and  only  the  foregoing 
very  general  directions  can  be  given  for  their  solution.  However, 
when  any  such  problem  admits  of  a  definite  solution,  it  will  be  found 
that  there  are  in  it  as  many  distinct  statements  as  unknown  numbers. 
These  distinct  statements  enable  us  to  express  all  the  unknown  num- 
bers in  terms  of  one  of  them.  The  algebraic  form  of  the  final  state- 
ment is  an  equation  in  this  one  unknown. 

ILLUSTRATIVE  EXAMPLES 

1.  What  number  is  as  much  greater  than  10  as  it  is  less 
than  54? 

Solution. 

Let  X  =  the  required  number. 

Then,  a;  —  10  =  the  difference  between  the  required  number  and  10, 
and  54  —  a;  =  the  difference  between  54  and  the  required  number. 

Since  the  two  differences  are  equal, 

X  -  10  =  54  -  x.  (1) 

Transposing,  a:  -h  a:  =  54  -H  10.  (2) 

Combining,  2  a:  =  64.  (3) 

Therefore,  x  =  32.  (4) 

Hence,  the  required  number  is  32. 
Check.  32  -  10  =  54  -  32. 

2.  A  dealer  bought  500  oranges  in  two  lots ;  the  first 
lot  at  the  rate  of  2J  cents  apiece,  and  the  second  at  the 
rate  of  2  cents  apiece.  He  sold  them  all  at  the  rate  of 
30  cents  a  dozen  and  gained  $2.25.  How  many  did  he 
buy  at  each  price? 

Solution. 

Let  X  =  the  number  of  oranges  in  the  first  lot. 

Then,  500  —  x  =  the  number  of  oranges  in  the  second  lot. 
Then,  f  a?  =  the  number  of  cents  in  the  cost  of  the  first  lot. 

and  2(500  —  x)=  the  number  of  cents  in  the  cost  of  the  second  lot. 
.*.  I"  a;  -f  2(500  —  x)—  the  number  of  cents  in  the  cost  of  both  lots. 
^-^^  X  30  =  the  number  of  cents  in  the  selling  price  of  both  lots. 


SIMPLE  EQUATIONS  89 

Then,  I  a:  -f-  2(500  -  x)+  225  =  ^^-  x  30.  (1) 

Simplifying,  |  a:  +  1000  -  2  x  +  225  =  1250.  (2) 

Transposing  and  combining,         J  a:  =  25.  (3) 

Whence,  x  =  50, 

and  500 -a:  =  450.  (4) 

Therefore,  the  dealer  bought  50  oranges  at  2-|  cents  apiece  and  450  at 
2  cents  apiece. 

Check.    I  X  50+2(500  -  50)  +  225  =  -^^^-  x  30. 

That  is,  1250  =  1250. 

3.  A  man  traveled  30  mi.  in  6  hr.  40  min.,  walking 
part  of  the  distance  at  the  rate  of  3  mi.  an  hour  and  rid- 
ing the  remaining  distance  at  the  rate  of  6  mi.  an  hour. 
How  far  did  he  walk? 

Solution.     Let  x  =  the  number  of  miles  he  walked. 

Then,  30  —  a;  =  the  number  of  miles  he  rode. 

Also,  -  =  the  number  of  hours  he  walked, 

and  =  the  number  of  hours  he  rode. 

6 

rvx.      t  a;  ,  30-  a:      20  ,,v 

Therefore,  -  +  —^—  =  —•  (1) 

Multiplying  both  members  of  (1)  by  6, 

2  a:  +  30  -  a:  =  40.  (2) 

Combining,  x  +  30  =  40.  (3) 

Whence,  x  =  10.  (4) 

Cheok.  10     3_0-ip^20.  ^ 

3  6  3  ^  ^ 

That  is,  \^  =  ^^-  (6) 

4.  A  number  is  composed  of  two  digits ;  the  digit  in 
the  tens'  place  is  one  more  than  twice  that  in  the  units' 
place,  and  if  36  is  subtracted  from  the  number  the  result- 
ing number  is  expressed  by  the  same  two  digits  taken  in 
the  reverse  order.     Find  the  number. 


90  ELEMENTARY  ALGEBRA 

Solution.         Let  x  =  the  units'  digit. 

Then,  (2  a:  -h  1)  =  the  tens'  digit. 

Also,  10(2  X  +  l)  +  a:  =  the  number, 
and        10  a:  +  (2  0?  +  1)  =  the  number  obtained  by  writing  the  digits 

in  the  reverse  order. 

By  the  conditions  of  the  problem, 

10(2a:+  l)+a:-36  =  10a:+(2a:+  1).  (1) 

Simplifying  (1),    9  a:  =  27.  '  (2) 

Dividing  by  9,  a;  =  3,  (3) 

and  2  x  + 1  =  7.  (4) 

Therefore,  the  required  number  is  73. 

EXERCISE  34 

1.  If  X  represents  a  certain  number,  what  represents 
the  number  increased  by  3? 

2.  What  number  increased  by  3  is  equal  to  15? 

3.  If  X  represents  a  certain  number,  what  represents 
the  number  diminished  by  2? 

4.  What  number  diminished  by  2  is  equal  to  10  ? 

5.  If  X  represents  a  certain  number,  what  represents 
four  times  the  number  ? 

6.  If  four  times  a  certain  number  is  30,  what  is  the 
number  ? 

7.  If  X  represents  a  certain  number,  what  represents  \ 
of  the  number?    What  represents  |  of  the  number? 

8.  If  I  of  a  certain  number  is  2J,  what  is  the  number  ? 

9.  A  certain  can  filled  with  lard  weighs  42  lb. ;  if  the 
can  weighs  4  lb.,  what  is  the  weight  of  the  lard? 

10.  Two  boys  have  36  cents ;  if  one  of  them  has  three 
times  as  much  as  the  other,  how  much  has  each? 

11.  Two  men  bought  100  fruit  trees;  if  one  of  them 
bought  10  more  than  the  other,  how  many  did  each  buy? 


SIMPLE  EQUATIONS  91 

12.  If  the  sum  of  two  angles  is  90°  and  one  of  them  is 
20°,  what  is  the  other? 

13.  If  the  sum  of  two  consecutive  integers  is  21,  what 
are  the  numbers? 

14.  The  sum  of  two  numbers  is  276.  One  of  them  is 
five  times  the  other ;  what  are  the  numbers  ? 

15.  The  number  of  pupils  in  a  certain  school  is  227  and 
the  number  of  girls  exceeds  the  number  of  boys  by  21. 
How  many  boys  are  there? 

16.  I  paid  8150  for  two  cows,  one  costing  §30  more 
than  the  other.     What  was  the  price  of  each? 

17.  The  difference  between  two  numbers  is  7  and  their 
sum  is  31.     What  are  the  numbers  ? 

18.  If  from  five  times  a  number  21  is  subtracted,  the 

remainder  is  9.     What  is  the  number? 

19.  Separate  24  into  two  parts  so  that  one  part  may  be 
equal  to  three  fifths  of  the  other. 

20.  A  woman  bought  a  certain  number  of  yards  of  dress 
goods  and  one  half  as  many  yards  of  lining.  If  she  bought 
24  yards  of  cloth,  how  many  yards  of  each  did  she  buy? 

21.  If  one  half  of  a  number  added  to  one  fourth  of  the 
number  is  7|^,  what  is  the  number  ? 

22.  Find  a  number  which  when  100  is  added  to  it  will 
give  a  result  equal  to  five  times  the  number. 

23.  Two  dealers  together  bought  15,000  bushels  of 
wheat,  one  of  them  buying  three  times  as  many  bushels 
as  the  other.     How  many  bushels  did  each  buy  ? 

24.  A  wagon  loaded  with  wheat  weighed  6390  lb.  If 
the  wagon  weighed  one  half  as  much  as  the  wheat,  what 
was  the  weight  of  the  wheat  ? 


92  ELEMENTARY  ALGEBRA 

25.  Three  times  a  certain  number  is  24  more  than  J  of 
the  number.     What  is  the  number  ? 

26.  The  result  of  subtracting  96  from  a  certain  number 
is  the  same  as  the  result  of  dividing  the  same  number  by 
13.     What  is  the  number  ? 

27.  The  difference  of  two  numbers  is  24  and  the  smaller 
is  I  of  the  larger.     What  are  the  numbers  ? 

28.  A  and  B  together  own  466  acres  of  woodland.  If 
22  times  A's  share  is  6  acres  less  than  B's  share,  how  much 
does  each  own? 

29.  A  dealer  sold  an  article  for  il2,  which  was  at  a 
gain  of  ^  of  the  cost.     What  was  the  cost? 

30.  A  dealer  sold  an  article  for  $12,  which  was  at  a  loss 
of  ^  of  the  cost.     What  was  the  cost? 

31.  The  wages  of  a  man  and  his  son  for  one  month  were 
il20.  If  the  son's  wages  were  |  of  the  father's,  what 
were  the  wages  of  each? 

32.  Find  three  consecutive  numbers  whose  sum  is  42. 

33.  The  sum  of  three  angles,  A,  B,  C,  is  180°.  If  B  is 
two  times  0  and  A  three  times  (7,  how  many  degrees  are 
there  in  each? 

34.  The  sum  of  the  angles  of  any  plane  triangle  is  180°. 
If  in  a  triangle  ABC^  angle  A  is  twice  angle  B  and  angle 
(7  is  J  of  angle  -B,  how  many  degrees  are  there  in  each 
angle? 

35.  A  storekeeper  found  that  he  had  #6.50  in  dimes 
and  quarters.  How  many  had  he  of  each  if  the  number 
of  coins  of  both  kinds  that  he  had  was  35? 

36.  A  man  wishes  to  divide  a  straight  line  40  ft.  long 
into  three  parts  so  that  the  first  part  may  be  4  ft.  less 
than  the  second  and  the  second  7  ft.  more  than  the  third. 
Required  the  length  of  each  part. 


SIMPLE  EQUATIONS  93 

37.  If  ^  of  a  pole  is  in  mud,  ^  of  it  in  water,  and  the 
remainder,  15  ft.  of  it,  above  water,  what  is  the  length  of 
the  pole  ? 

38.  A  baseball  team  won  63  games,  which  were  |  of  the 
games  that  it  played.     How  many  games  did  it  play? 

39.  In  sorting  melons  27  less  than  |  of  them  were 
found  to  be  defective.  If  45  of  the  melons  were  found  to 
be  in  good  condition,  how  many  of  them  were  defective? 

40.  A  man  sold  3  acres  more  than  ^  of  his  lot  and  had 
2  acres  less  than  half  of  it  left.  Find  the  number  of 
acres  in  the  lot. 

41.  If  a;  represents  the  number  of  dollars  in  the  cost  of 
an  article,  what  represents  the  number  of  dollars  in  the 
gain  if  the  rate  of  gain  is 

50%?     25%?     20%?     100%?     121%?     62i%? 

50  1  X 

Suggestion.       50%  of  a:  =  — —  x  =  -x,  or  -■ 

42.  If  X  represents  the  number  of  dollars  in  the  cost  of 
an  article,  what  represents  the  number  of  dollars  in  the 
loss  if  the  rate  of  loss  is 

5%?     4%?      75%?      371%?      331%?      6i%? 

43.  If  X  represents  the  number  of  dollars  in  the  cost  of 
an  article,  what  represents  the  number  of  dollars  in  the 
selling  price  if  the  rate  of  gain  is 

25%?         30%?         80%?         621%?         200%? 

44.  If  X  represents  the  number  of  dollars  in  the  cost  of 
an  article,  what  represents  the  number  of  dollars  in  the 
selling  price  if  the  rate  of  loss  is 

20%?         10%?         121%?         16f%?         8|%? 

45.  A  dealer  gained  25  %  by  selling  a  coat  at  a  profit  of 
$51.     Find  the  cost  of  the  coat. 


94  ELEMENTARY  ALGEBRA 

46.  Some  lemons  were  sold  at  a  loss  of  6  cents  a  dozen. 
If  the  rate  of  loss  was  16|%,  what  was  the  cost  ? 

47.  A  farmer  sold  a  horse  for  §220,  which  was  at  a 
gain  of  10  %.     What  was  the  cost  ? 

48.  A  used  automobile  was  sold  for  §600,  which  was 
20  %  less  than  cost.     What  was  the  cost  ? 

49.  The  difference  between  two  numbers  is  328,  and  the 
larger  is  42  times  the  smaller.     Find  one  of  the  numbers. 

50.  A  tennis  court  for  doubles  is  42  ft.  longer  than  its 
breadth.  The  distance  around  the  court  is  228  ft.  Find 
the  length  and  breadth  of  the  court. 

51.  An  acre  of  wheat  yielded  25,000  lb.  more  straw 
than  grain.  The  weight  of  the  grain  was  f  of  the 
weight  of  the  straw.     What  was  the  weight  of  tlie  grain? 

52.  A  man  bequeathed  §45,000  to  his  wife,  daughter, 
and  son.  The  daughter  received  §5000  more  than  the 
son,  and  the  wife  received  three  times  as  much  as  the  son. 
How  much  did  each  receive  ? 

53.  A  man  is  27  years  older  than  his  son ;  12  years 
hence  he  will  be  twice  as  old  as  his  son  will  be  then. 
How  old  is  his  son  ? 

54.  Divide  §30  among  three  persons  so  that  the  first 
person  shall  receive  three  times  as  much  as  the  second, 
and  the  third  person  §5  more  than  the  second. 

55.  Part  of  a  sum  of  §3000  was  invested  at  4%  and 
the  remainder  at  4|  %  ;  the  total  income  from  these  invest- 
ments was  §126.25.    How  much  was  invested  at  each  rate? 

56.  Divide  68  into  two  parts  so  that  one  third  of  one 
part  may  equal  one  fourth  of  the  other. 

57.  A  man  is  60  years  old  and  his  son  is  30  ;  how  many 
years  ago  was  the  man  just  three  times  as  old  as  his  son  ? 


SIMPLE  EQUATIONS  95 

58.  What  number  diminished  by  ^  of  itself  equals  1  less 
than  I  of  itself  ? 

59.  All  school  buildings  should  have   the   total   light 
space  equal  to  at  least  20  %  of  the  floor  space  ;  what,  then," 
is  the  greatest  amount  of  floor  space  that  a  schoolroom 
should  have  whose  light  space  is  180  sq.  ft.  ? 

60.  A  straight  line  is  divided  into  two  parts,  one  of 
which  is  30  in.  longer  than  the  other.  Seven  times  the 
shorter  piece  equals  two  times  the  longer.  How  long  is 
the  line  ? 

61.  A  has  $3  more  than  B,  and  B  has  §6  more  than  C  ; 
together  they  have  #111.     How  much  has  each? 

62.  On  a  farm  there  are  twice  as  many  turkeys  as  there 
are  dugks,  and  five  times  as  many  chickens  as  there  are 
turkeys.  There  are  260  of  the  three  kinds  in  all.  How 
many  are  there  of  each  ? 

63.  How  many  pounds  of  tea  at  40  ct.  a  pound  must  be 
mixed  with  20  lb.  at  75  ct.,  in  order  that  the  mixture  may 
be  worth  50  ct.  a  pound  ? 

64.  The  perimeter  of  a  rectangle  is  1000  yd.  and  its 
altitude  is  four  times  its  base.  What  is  the  length  of 
the  base  ? 

65..  Eight  men  hired  a  yacht,  but  by  taking  in  four 
more  the  expense  of  each  was  diminished  f  1 ;  how  much 
did  they  pay  ? 

66.  A  man  bought  2-cent  stamps,  5-cent  stamps,  and 
11-cent  stamps,  of  each  the  same  number  ;  if  he  paid  72  ^ 
for  the  lot,  how  many  of  each  did  he  buy  ? 

67.  A  man  saved  §1350  in  three  years.  He  saved  twice 
as  much  the  second  year  as  the  first,  and  three  times  as 
much  the  third  as  the  second.  How  much  did  he  save 
the  first  year  ? 


96  ELEMENTARY  ALGEBRA 

68.  A  man  spends  ^  of  his  yearly  income  for  board  and 
lodging,  I  of  the  remainder  for  clothes  and  other  expenses, 
and  saves  ^500  a  year.     What  is  his  income  ? 

69.  What  number  increased  by  |  of  itself  equals  the 
sum  of  I  of  the  number  and  9  ? 

70.  A  man  invests  f  of  his  capital  at  5%,  and  the  re- 
mainder of  it  at  4|  % ;  his  annual  income  from  both  in- 
vestments is  i  240.     Find  his  capital. 

71.  "  Eight  years  ago,"  said  a  man  to  his  son,  "  I  was 
thirteen  times  as  old  as  you  were,  and  four  years  hence, 
I  shall,  if  I  live,  be  four  times  as  old  as  you  will  be  then.'* 
What  is  the  man's  age  ? 

72.  A  merchant  mixes  30  lb.  of  tea  which  cost  40  ct.  a 
pound,  and  20  lb.  which  cost  60  ct.  a  pound.  What  is 
the  mixture  worth  per  pound  ? 

73.  If  I  spend  $70  for  rugs  and  $S6  for  chairs,  and  then 
have  left  one  fourth  of  what  I  had  at  first,  how  much  have 
I  remaining? 

74.  A  dealer  has  coffee,  some  at  20  ct.  and  some  at  35  ct. 
per  pound ;  he  wishes  to  make  a  mixture  of  100  pounds 
which  shall  be  worth  30  ct.  a  pound.  How  many  pounds 
of  each  must  he  use  ? 

75.  A  train  ran  from  Pittsburgh  to  Philadelphia  in  7^ 
hours ;  if  it  had  traveled  10  miles  an  hour  slower,  it  would 
have  taken  10  hours.  Find  the  distance  from  Pittsburgh 
to  Philadelphia. 

76.  It  is  required  to  find  a  number  such  that  if  it  be 
multiplied  by  3  and  the  product  increased  by  7,  the  result 
shall  be  the  same  as  if  it  were  increased  by  8,  and  the  sura 
multiplied  by  2. 


SIMPLE  EQUATIONS  97 

77.  10  lb.  of  tea  and  12  lb.  of  coffee  together  cost  19.60. 
If  a  pound  of  tea  cost  30  ct.  more  than  a  pound  of  coffee, 
find  the  cost  per  pound  of  each. 

78.  A  packer,  engaged  to  pack  500  tumblers,  received 
8  ct.  for  every  one  that  arrived  at  its  destination  in 
good  condition,  and  forfeited  15  ct.  for  every  one  broken. 
He  received  il7.34.     How  many  were  broken  ? 

79.  A  cask  contains  a  mixture  of  25  gallons  of  vinegar 
and  5  gallons  of  water ;  a  certain  quantity  is  drawn  out 
and  replaced  by  water  and  then  the  mixture  consists  of 
10  gallons  of  vinegar  with  20  gallons  of  water.  How 
many  gallons  were  drawn  out? 

Suggestion.  If  x  represents  the  number  of  gallons  drawn  out ; 
then  I  X  represents  the  number  of  gallons  of  vinegar  drawn  out,  and 
25  —  I  a;  represents  the  number  of  gallons  of  vinegar  left  in. 

80.  A  bottle  contains  a  mixture  of  1  pint  of  cream  and 
3  pints  of  milk ;  a  certain  quantity  is  removed  and  re- 
placed by  milk,  and  then  the  mixture  contains  J  of  a  pint 
of  cream.     How  much  was  removed  ? 

81.  I  traveled  22  mi.  in  8  hr.,  walking  part  of  the 
way  at  4  mi.  per  hour,  and  riding  the  rest  of  it  at  10  mi. 
per  hour.     How  far  did  I  walk  ? 

82.  A  person  wishing  to  give  50  cents  apiece  to  some 
boys,  finds  that  he  has  not  money  enough  by  25  cents  ;  but 
if  he  gives  them  40  cents  apiece  he  will  have  35  cents  re- 
maining.    Required  the  number  of  boys. 

83.  A  workman  received  ^2.50  and  his  board  for  each 
day  that  he  worked,  and  paid  60  ct.  for  board  for  each 
day  that  he  did  not  work.  For  90  da.  he  received  $132 ; 
how  many  of  these  days  did  he  work  ? 


98  ELEMENTARY  ALGEBRA 

84.  It  is  required  to  find  two  numbers  whose  sum  is 
12,  such  that  if  ^  the  less  be  added  to  J  the  greater,  the  sum 
shall  be  equal  to  ^  the  greater  diminished  by  ^  the  less. 

85.  How  many  pounds  of  water  must  be  added  to  40  lb. 
of  a  5  %  solution  of  salt  to  obtain  a  4  %  solution  ? 

86.  How  many  pounds  of  salt  must  be  added  to  80  lb. 
of  a  10%  solution  of  salt  to  obtain  an  11|  %  solution  ? 

87.  A  certain  number  consists  of  two  digits,  the  one  in 
the  units'  place  being  twice  that  in  the  tens'  place.  If  18 
be  added  to  the  number,  the  resulting  number  is  repre- 
sented by  the  same  digits  reversed.  What  is  the  original 
number  ? 

88.  A  certain  number  consists  of  two  digits,  the  one  in 
the  units'  place  being  three  times  the  one  in  the  tens' 
place.  If  the  order  of  the  digits  be  reversed  and  16  be 
added  to  the  resulting  number,  the  new  number  will  be 
three  times  the  original  number.  What  is  the  original 
number  ? 

89.  Into  what  two  sums  can  $2700  be  divided  so  that 
the  income  from  one  at  5  %  shall  equal  the  income  from 
the  other  at  4  %  ? 

90.  M's  income  is  $500  a  year  more  then  N's  and  each 
saves  ^  of  his  income.  At  the  end  of  10  years  M  has  saved 
1|  times  as  much  as  N.    What  is  the  yearly  income  of  each  ? 

91.  A  certain  medicine  contains  80%  alcohol.  How 
much  water  must  be  added  to  1  quart  of  it  so  that  the 
mixture  shall  contain  only  10  %  alcohol  ? 

92.  During  one  year  a  traction  company  carried 
3,000,000  fewer  passengers  than  in  the  preceding  year ; 
but,  as  the  average  fare  had  been  raised  from  4.1  ct.  to 
5.2  ct.,  the  receipts  were  $394,000  more.  How  many 
were  carried  in  each  year  ? 


SIMPLE  EQUATIONS  99 

93.  If  one  machine  can  grind  10  bu.  of  grain  in  2|  hr. 
and  another  can  grind  10  bu.  in  IJ  hr.,  how  long  will  it 
take  both  machines  to  grind  100  bu.  of  grain  ? 

94.  The  digit  in  the  units'  place  of  a  number  composed 
of  two  digits  is  4  less  than  3  times  that  in  the  tens'  place; 
the  sum  of  the  digits  plus  27  is  equal  to  the  number. 
Find  the  number. 

95.  Twice  the  digit  in  the  tens'  place  of  a  number 
composed  of  two  digits  is  7  greater  than  that  in  the  units' 
place ;  if  7  is  subtracted  from  the  number,  the  remainder 
is  5  times  the  sum  of  the  two  digits.     Find  the  number. 

96.  If  one  machine  can  skim  75  gal.  of  milk  in 
1|  hr.  and  another  60  gal.  per  hour,  how  long  must  both 
run  to  skim  300  gal.  of  milk  ? 

97.  A  man  invested  $  5500  in  two  business  enterprises. 
On  the  first  investment  he  lost  6  %  and  on  the  second  he 
gained  5%.  His  net  gain  was  $55.  How  many  dollars 
did  he  invest  in  each  enterprise  ? 

98.  There  is  a  reservoir  which  can  be  supplied  with 
water  from  three  different  inlets  ;  from  the  first  it  can  be 
filled  in  12  hr.,  from  the  second  in  18  hr.,  and  from  the 
third  in  36  hr.  In  what  time  will  it  be  filled  if  it  is 
being  supplied  from  all  inlets  at  the  same  time  ? 

99.  A  certain  principal  will  earn  120  more  interest  in 
8  yr.  at  6%  than  it  will  in  ^  yr.  at  5%.  What  is  the 
principal  ? 

100.  A  certain  principal  will  in  4  yr.  at  5  %  amount  to 
$10  less  than  the  same  principal  will  amount  to  in  5  yr. 
at  4|-%.     What  is  the  principal? 

101.  The  sum  of  two  numbers  is  30  and  one  of  them  is 
6  less  than  the  other.     Find  the  numbers. 


CHAPTER  IV 

TYPE   PRODUCTS   AND   FACTORS 

79.  Rational  operations.  Addition,  subtraction,  multi- 
plication, and  division  are  called  the  rational  operations  of 
algebra. 

80.  Rational  expression.  An  expression  which  in- 
volves only  rational  operations  is  called  a  rational  ex- 
pression. 

81.  Integral  expression  with  respect  to  any  letter.  An 
expression  is  said  to  be  integral  with  respect  to  any  letter 
when  it  does  not  involve  a  division  either  by  that  letter 
or  by  a  polynomial  containing  that  letter. 

Thus,  a^  —  2a  -h  -  is  integral,  but  -  and  '^—^  are  not  integral, 

..,  ,.  3  a  2a— 1 

with  respect  to  a. 

82.  Integral  expression.  An  integral  expression  is  an 
expression  that  is  integral  with  respect  to  each  one  of  the 
letters  which  it  contains. 

Thus,  a^h  —  Xxy  -\-  2  is  an  integi-al  expression. 

83.  Degree  of  a  monomial.  The  degree  of  an  integral 
monomial  is  equal  to  the  number  of  its  literal  factors. 

Thus,  3  x^y  is  a  monomial  of  the  third  degree. 

84.  Degree  of  an  expression.  The  degree  of  an  integral 
algebraic  expression  is  the  same  as  the  degree  of  the  term 
or  terms  of  the  expression  which  are  of  the  highest 
degree. 

100 


TYPE  PRODUCTS  AND  FACIO^RS  I'^i 

Thus,  Sa%  +  2  abc  —  5  a  is  an  algebraic  expression  of  the  third 
degree,  and  a:^  +  5  a:  +  6  is  one  of  the  second  degree. 

Note.  When  all  the  terms  of  an  expression  are  of  the  same 
degree,  the  expression  is  called  homogeneous. 

Thu^,  3  aH)  +  2  h^c  —  5  c%  is  a  homogeneous  expression. 

85.  Degree  of  an  expression  with  respect  to  a  particular 
letter.  The  degree  of  an  integral  expression  with  respect 
to  a  particular  letter  is  the  same  as  the  index  of  the 
highest  power  of  that  letter  in  the  expression. 

Remark.  It  is  convenient  to  classify  expressions  according  to 
their  degree  with  respect  to  a  given  letter. 

Thus,  with  respect  to  x  : 

ax  -\-  b  is  a  linear  expression,  or  an  expression  of  the  Jirst  degree; 

ax^  4-  &x  +  c  is  a  quadratic  expression,  or  an  expression  of  the 
second  degree  ; 

ax^  +  bx^  -]-  ex  +  d  is  a.  cubic  expression,  or  an  expression  of  the 
third  degree; 

ax*  +  bx^  +  cx^  4-  dx  -\-  e  is  an  expression  of  the  fourth  degree. 

EXERCISE  36 

1.  State  the  degree  of  each  of  the  following  monomials: 

3  X1/Z  ;    —  2  0^1/  ;   1  a^;    —  3  ci^i/h. 

2.  State  the  degrees  with  respect  to  x  of  each  of  the 
monomials  in  example  1. 

3.  State  the  degree  of  the  following  expressions  : 
2x-\-S;  ax  — by;  aa^-^-bx+e;  a^  —  y^;  x^yH'^ -^2xyz —  Z. 

4.  State  the  degrees  with  respect  to  x  of  each  of  the 
expressions  in  the  preceding  exercise. 

5.  Which  of  the  following  expressions  are  integral  with 
respect  to  each  of  the  letters  contained? 

o^25        3^2  _  5^     2a:2  1     a^^hy^ 

3  3  i  + 1        a  2  ab 


102  STLEMENTARY  ALGEBRA 

6.  State  the  degree  of  each  of  the  following  products : 

7.  State  the  degree  of  each  of  the  following  quotients : 
3a6 -5-2^2.  a253^2a52;  Sx'-^'Sx;  y^yH^~xy^^, 

8.  Which  of  the  following  expressions  are  homogeneous  ? 
a^^1a-\-W'\  a  +  254-3c;  2^2^33,^.  ^_^_f_22. 

x  +  y  -\-l;  x^  -\- ip'  -\- z^  —  xyz  ;   he  +  ca  +  ah. 

9.  Write  a  homogeneous  polynomial  of  five  terms  using 
the  letters  a,  6,  and  c. 

10.  Write  two  homogeneous  polynomials  of  three  terms 
each,  find  their  product,  and  state  whether  or  not  it  is 
homogeneous. 

86.  The  square  of  a  monomial.  From  section  58,  we 
have, 

That  is,  (ic»»)2  =  a:2m^ 

Hence, 

Rule.  To  square  a  power  of  a  number^  multiply  its  ex- 
ponent hy  '2. 

By  the  commutative  law  of  multiplication,  section  56, 

That  is,  (arb'^y  =  (a'»)2(5»)2  =  a2m52n. 

Hence, 

Rule.  To  square  a  monomial^  multiply  the  exponent  of 
each  of  its  factors  by  2. 

Remark.  When  a  monomial  has  a  numerical  coefficient,  it  is 
usually  preferable  actually  to  square  the  coefficient  rather  than  to 
indicate  its  square. 

Thus,  (3  x^yzY  =  3\x^)Y^^  =  »  ^Y^^- 


TYPE  PRODUCTS  AND  FACTORS  103 

ILLUSTRATIVE  EXAMPLES 

1.  Square  a^. 

Solution.  (a^y  =  a\ 

2.  Square  6^. 

Solution.  (68'«)2  =  66«. 

3.  Square  —  e5  a^h^c. 

Solution.  ( -  5 a^b^cy  =  (-  5ya%^c^  =  2ba%*c^. 

4.  Square  (a  +  by(c  —  dy. 

Solution,     [(a  +  b)\c  -  dy^  =  («  +  ^)*  (^  "  ^Y- 

EXERCISE  36 

(Solve  as  many  as  possible  at  sight.) 
Find  the  square  of  each  of  the  following,  as  indicated  : 

1.  (-3)2;   (-2a)2;   (5a)2;   (-2x  32^)2. 

2.  [(a  +  J)P;   [2(«  +  5)]2;   [_3(a  +  6)P. 

3.  (  -  5  2^y ^5)2 ;  (-2.3  a63c2)2. 

4.  (-2a-)2;   (3a3-)2. 

5.  [2a(5  +  (?)P;   [-3a3(5  +  c?)4]2. 

6.  [|a]2;    [-|a]2;    [| .a^6]2. 

7.  [-fa2(6  +  ^)2(2;  +  ^)3]2. 

8.  Why  is  the  square  of  all  numbers  which  we  have 
considered  necessarily  positive  ? 

9.  Why  are  the  exponents  of  the  literal  factors  which 
occur  in  the  square  of  a  monomial  necessarily  even 
numbers  ? 

10.  If  the  numerical  coefficient  of  a  monomial  is  a  per- 
fect square,  and  the  exponents  of  the  literal  factors  are  all 
even  numbers,  what  can  be  said  of  the  monomial  ? 


104  ELEMENTARY  ALGEBRA 

87.   The  cube  of  a  monomiaL     From  section  58,  we  have 

That  is,  (af^y  =  a^. 

Hence, 

Rule.  To  cube  a  power  of  a  number  multiply  its  exponent 
byZ. 

By  the  commutative  law  of  multiplication,  section  56, 

That  is,  (iaH^y  =  (a^)3(5»)3  =  c^H^-. 

Hence, 

Rule.  To  cube  a  monomial  multiply  the  exponent  of  each 
of  its  factors  by  Z. 

Remark.  When  a  monomial  has  a^  numerical  coefficient,  it  is 
usually  preferable  to  cube  the  coefficient  rather  than  to  indicate  its 
cube. 

Thus,  (3  x^yzY  =  S^x^yf^^  =  27  xy^- 

ILLUSTRATIVE   EXAMPLES 

1.  Cubea2. 

Solution.  (a2)8  =  a^. 

2.  Cube  a^"*. 

Solution.  (x^y  =  a;9«. 

3.  Cube  -  5  flS^c?. 

Solution.  (  -  5  a%^cy  =  (  -  5y(a»y(byc»  =  -  125  a^'^cK 

4.  Cube  (a  4-  hy(c  -  dy. 

Solution,    [(a  +  by(c  -  dyf  =  (o  +  by(c  -  dy. 

EXERCISE  37 

(Solve  as  many  as  possible,  at  sight.) 
Find  the  cube  of  each  of  the  following,  as  indicated : 

1.  (2)8;   (-3)8;   (-2a)8;  (5  ay. 

2.  (abc^y;   (2ab^cyi   {-^a^yhy. 


TYPE  PRODUCTS  AND  FACTORS      105 

3.  [(a  +  b^Y;    [2(a  +  6)]3;  [_3(«+^)]=^. 

4.  (-2a^y;   (3a3'")3. 

5.  l2a(b-\-c)Y;   [-Sa^b  +  cyy, 

6.  [faP;   [-|aP;   [2a26]3. 

8.  When  is  the  cube  of  a  number  positive?    When 
negative  ? 

9.  Why  are  the  exponents  of  the  literal  factors  which 
occur  in  the  cube  of  a  monomial  necessarily  divisible  by  3? 

10.  If  the  numerical  coefficient  of  a  monomial  is  a 
perfect  cube  and  the  exponents  of  the  literal  factors  are  all 
multiples  of  3,  what  can  be  said  of  the  monomial  ? 

88.  Square  root  and  cube  root  of  a  monomial.  When 
the  square  of  a  number  a  is  equal  to  a  given  number 
A^  the  number  a  is  called  a  square  root  of  A.  Also, 
when  the  cube  of  a  number  a  is  equal  to  a  given  number 
A^  the  number  a  is  called  a  cube  root  of  A. 

Thus,  3  is  a  square  root  of  9  since  (3)^  is  equal  to  9.  Also,  3  is  the 
cube  root  of  27,  since  (3)^  is  equal  to  27. 

There  are  always  two  square  roots  of  a  number,  the  one 
being  positive  and  the  other  negative. 

Thus,  since  (+2)2  =  4  and  (-2)2  =  4,  both  +  2  and  -  2  are 
square  roots  of  4. 

Since  the  cube  of  a  positive  number  is  positive,  and  the 
cube  of  a  negative  number  is  negative,  it  follows  that  the 
cube  root  of  a  positive  number  is  positive  and  the  cube 
root  of  a  negative  number  is  negative. 

Thus,  (+2a)8  =  +  8a3  and  (-2a)8  =  -8a8;  therefore,  +2a 
is  the  cube  root  of  +  8  a*  and  -  2  a  is  the  cube  root  of  —  8  a*. 

Remark.  For  the  present,  a  number  will  be  considered  as  having 
but  one  cube  root.  Later  it  will  be  shown  that  there  are  three  dif- 
ferent expressions  which  when  cubed  give  the  same  number. 


106  ELEMENTARY  ALGEBRA 

89.  Notation.  The  radical  sign  V  is  used  in  algebra  to 
indicate  a  square  root ;  similarly,  V  is  used  to  indicate  a 
cube  root. 

Note  1.  In  such  expressions  as  Vah  fhe  sign  V  does  not  include 
h.  The  square  root  of  the  whole  expression  is  indicated  either  by 
y/(ab)  or  Vab,  the  vinculum  over  ah  serving  the  purpose  of  paren- 
theses. 

Note  2.     The  sign  ±  is  read  plus  or  minus. 

Thus,  y/9  =  ±  3  is  read  the  square  roof  of  9  is  equal  to  plus  or  minus  3. 
It  is  agreed,  however,  that  +  VO  shall  mean  +  3  and  —  V9  shall  mean 
-3. 

EXERCISE   38 

(Solve  as  many  as  possible  at  sight.) 

1.  Why  is  it  not  possible  to  find  a  numerical  value  of 
the  square  root  of  a  negative  number  ? 

2.  In  finding  a  square  root  of   a\  by   what  number 
must  the  exponent  be  divided  ? 

3.  In  finding  the  square  root  of  aP^^  by  what  number 
must  the  exponent  be  divided  ? 

4.  Give  a  rule  for  extracting  a  square  root  of  a  num- 
ber such  as  ic^m  [§  86]. 

5.  Give  a  rule  for  extracting  a  square  root  of  such  an 
expression  as  a^b^^  [§  86]. 

6.  How  can  the  result   obtained  by  taking  a  square 
root  of  a  number  be  checked  ? 

7.  In  finding  the  cube  root  of  a^  by  what  number  must 
the  exponent  be  divided  ? 

8.  In  finding  the  cube  root  of  a^"*,  by  what  number 
must  the  exponent  be  divided  ? 

9.  Give  a  rule  for  extracting  the  cube  root  of  a  num- 
ber such  as  a^""  [§  87]. 


TYPE  PRODUCTS  AND  FACTORS  107 

10.  Give  a  rule  for  extracting  the  cube  root  of  such  an 
expression  as  a^b^"'  [§  87]. 

11.  How  can  the  cube  root  of  a  number  be  checked  ? 
Find  the  following  roots,  as  indicated : 

12.  V^;   </'^;   </S^;  Vf^^p.   ^8^356.   VT6^*P. 

13.  V9"^^;   V36^y^. 


14.    </21a%^;   V-27 
3 


15.  V25(a  +  5)2;   V-125(aH-5)3. 

16.  Vf  dXb  +  ey  ;   ^2/«3(5  +  ^)3. 

18.    >/>-1000(a  + 2^)9(2  J -3)5*. 

Type  Products 

90.  Certain  algebraic  identities  which  occur  in  multi- 
plication are  specially  important  owing  to  their  frequent 
occurrence.  They  serve  as  models  for  other  multiplica- 
tions, and  for  this  reason  should  be  memorized. 

91.  The  distributive  law  [§  60]. 

a(b-\-c}  =  ab-{-ac.  (1) 

a(b-c)=ab-ac.  (2) 

ILLUSTRATIVE  EXAMPLES 

1.  x(^y  —  z)  =  xy  —  xz. 

2.  12(3a;^-^/)=36a;  +  12^/. 

3.  —  3  x^y  (a:  —  2  y)  =  —  3  oi^y  -f  6  a^y . 

4.  2y{x-y^z)  =  2yx  —  '2y{y  +  z)  =  2yx  —  2y'^-2yz. 

EXERCISE  39 

Multiply : 

1.    _2(a-4).  2.    Zx(2x-'^y). 

3.    |a(J6-c)-  *•    2wv(2w-3v). 


108  ELEMENTARY  ALGEBRA 

5.  3a6(3a-2J).  6.  |(3a:-6y). 

7.  —  xy  {x^  —  y)'  8.  (5  —  a)  d^, 

9.  (- 2a;+ 3)(-2a;).  10.  a{h-\-c  +  d). 

11.  f(52-|-J52).  12.  (3a:2^-52)(-3a;?/«2). 

€> 

13.  lm{l  +  m-\-l).  14.  2a;(i/-f2-0- 

15.  (2x-'6y^-bz){-2x).  16.  3a(5  -  <?  + ^ -/). 

17.  (-2a5)(-2a  +  35).  18.  (2  abc  -  1  bcd^- hd), 

19.  -  3^2^(11  t;2- 7 «3).  20.  (a-5  +  2c)(-3  5(?). 

21.  ^Iffi^fq-b^f),  22.  \'KH(W'^}P'-^Bh) 

92.  The  square  of  a  binomial.      By  multiplication  we 

find  that 

(a4-&)2  =  a2  +  2a&+62.  (1) 

This  identity  may  be  expressed  in  words  as  follows : 

The  square,  of  the  sum  of  two  numbers  is  the  square  of  the 
first  plus  twice  their  product^  plus  the  square  of  the  second. 

Replacing  ^  by  —b  in  identity  (1),  we  have  [§  67 
note] : 

[a  +(-  5)]2=  a2  +  2  a(-  5)4-(-  6)2.     Performing  the 

indicated  operations, 

(a-6)2  =  a2_2a&+fi2,  (2) 

This  identity  may  be  expressed  in  words  as  follows : 
The  square  of  the  difference  of  two  numbers  is  the  square  of 

the  first  minus  twice  their  product^  plus  the  square  of  the 

second, 

ILLUSTRATIVE  EXAMPLES 

1.    Square  35  by  use  of  identity  (1.). 

Solution.  35^  =  (30  +  5)2 

=  30^  +  2  X  5  X  30  4  5« 
=  900  +  300  +  2;-)  ^-  1225. 


Leonard  Euler  (1707-1783)  was  born  at  Bale  and  died  at 
St.  Petersburg.  He  wrote  on  almost  all  the  branches  of  mathe- 
matics then  known,  revising  almost  all  those  of  pure  mathematics. 
In  1770  he  published  an  algebra  at  St.  Petersburg  which  was 
translated  into  French  in  1794  by  the  celebrated  mathematician 
Lagrange. 


TYPE  PRODUCTS  AND  FACTORS      109 

2.  Square  19  by  use  of  identity  (2). 
Solution.  19''=(20-1)2 

=  20^  -  2  X  20  X  1  +  1* 
=  400  -  40  +  1  =  361. 

3.  Square  2  a  +  3  6. 

Solution.     (2  a  +  3  6)2  =  (2  a)^  +  2(2  a) (3  b)  +  (3  by 

=  4  a2  +  12  a6  +  9  &2. 

4.  Square  x^  —  2. 

Solution.         (xy  -  2)2  =  (xyY  -  2(xy) (2)  +  (2)« 
=  a;2y2  -  4  a:y  +  4. 

5.  Square  |  a;  —  f  y. 

Solution,     (f.:  -^yy=(ixy  _2(|a:)(f  i/)  +  (f  y)2 
=  Aa;2-  4a,y+  _9^y2. 


BXEBCISE  40 

Write    the    squares 

of    the    following 

binomials    as 

indicated : 

1.    (2:4-1)2. 

2. 

(2  0.-^)2. 

3. 

(a -3  5)2. 

4.    (2(?  +  3(^)2 

5. 

(7  a -2)2. 

6. 

(3  5  +  2)2. 

7.    (5«-3  6)2. 

8. 

(5cd-S  ay. 

9. 

(a +  1)2. 

10.    (2  a; -1)2. 

11. 

(Sax-h2bi/y. 

12. 

(2  a; -9)2. 

13.    (a  +  i)2. 

14. 

(3  m  -  2  71)2. 

15. 

Cal  +  sy. 

16.    (3  a: +  4)2. 

17. 

(5  a: -3)2. 

18. 

(2mn-S  w)2. 

19.    (abc-^iy. 

20. 

(xi/z-iy. 

21. 

3P. 

22.  nl 

23. 

m\ 

24. 

78l 

25.    29'. 

26. 

99^ 

27. 

(15:^  +  2^)2. 

28.    (13a- 3^)2. 

29. 

a^-i^)'. 

30. 

Cix-izy. 

31.    (fa; -1^)2. 

32. 

(la -1)2. 

33. 

(1  7/1 +  2)2. 

34.    Is  9  x^  -\-  4:  y^  -^  6  XT/  a  perfect  square ;  that  is,  the 
square  of  a  binomial? 


no  ELEMENTARY  ALGEBRA 

35.    Give   a   rule   for   determining   whether   or   not   a 
trinomial  is  a  perfect  square. 

Which  of  the  following  trinomials  are  perfect  squares  ? 

36.  ar^  +  2a:  +  4.  37.  jt?2  +  6  jo -f  9. 

38.  m^  +  n^-2  mn,  39.  ^x^  +  l  +  ^x. 

40.  9  2^  +  4  +  6  a;.  41.  a;^  +  4  y2  _  4  ^^^ 

42.  a^-xi-l.  43.  4m2  +  25-20m. 

44.  4:a^-\-4:x-l,  45.  r2-2r8-«2. 

46.  l-2jp+jp2.  47.  4-12w  +  3m2. 

93.  The  square  of  a  trinomiaL 

Since  a  -^  b  -\-  e  =  a  -{-  (^b  -\-  c}, 

(^a  +  b-^cy  =  la^(b-{-c}y 

=  a^^  2  aQb  +  c)-h(b  +  cy 
=:a^+2ab  +  2ac-\-b^-h2bc+c\ 

Hence, 

(ia  +  b+cy  =  (f-{-b^+c^-\-2ab  +  2ac-{-2bc. 

This  identity  may  be  expressed  in  words  as  follows  : 
The  square  of  a  trinomial  is  equal  to  the  sum  of  the  squares 

of  its  terms  plus  twice  the  sum  of  the  products  of  all  pairs 

of  the  terms. 

Note.     In  a  trinomial  there  are  three  pairs  of  terms. 
Thus,  in  the  trinomial  2  x  —3  y  -\-  dz  the  three  pairs  of  terms  are 
2  X  and  —  3  y,2  x  and  6z,  —  S  y  and  5  z. 

ILLUSTRATIVE  EXAMPLES 

1.  Square  x-\-y  —  z. 

Solution.     {x-{-y-zy  =  x'^-Vy^-\-{-zy  +  2xy-\-2x{-z)  +  2y{-z^ 
=  x^+  y^  -\-  z^-{-2xy  -2xz-2  yz. 

2.  Square   2  a;  —  3  ^  —  1    and    check ;    let   x  =  2    and 

y  =  i. 


TYPE  PRODUCTS  AND  FACTORS  111 

Solution.     (2x  -Sy  -ly 
=  (2xy+(-Syy+(-iy  +  2(2x)(-Sy)-]-2(2xX-l)+2(-^yX-l) 
=  4:  x^  +  Q  y^  +  1  -  12  xy  -4:X+Qij. 

Check.  *(2x-3y-iy  =  4:X^+9y^  +  l-12xy-4:X-\-6y 
(4  _  3  -  1)2=  16  +  9  +  1-24-8+6 
0  =  0. 

EXERCISE  41 

Square  the  following  trinomials,  as  indicated,  and  check 
by  substituting  a  =  1,  ^  =  2,  <?  =  3,  a:  =  1,  «/  =  —  1,  ^  =  2. 
1.    (a;  +  ^ 4-  2)2.  2.    (x-y  -  zy. 

3.    (a -+5 +  1)2.'  4.    (a  +  5+2;)2. 

5.    (x-\-2y  +  zy.  6.    (ia-x-\-^e)\ 

7.    (2  a  -  3  ^  -  1)2.  a    (3  a:  -  2  1/  -  2)2. 

9.    (a +  2  6-3)2.  10.    {yz -\- zx  ^- xyy. 

11.    (a2  +  «  +  i)2,  12.    (x^-xy^y'^y. 

94.  The  square  of  a  polynomial.  When  a  polynomial 
contains  more  than  three  terms,  it  may  be  expressed  as  a 
binomial  or  a  trinomial  by  the  use  of  parentheses. 

Thus,  the  polynomial  2  a  +  3  />  —  c  +  5  rZ  may  be  written, 
as  a  binomial,  (2a  +  3^)  +  (— c  +  5rf); 

as  a  trinomial,  (2  a  +  3  &)  —  c  +  5  c?. 

By  repeated  application  of  the  rules  for  finding  the 
square  of  a  binomial  or  a  trinomial,  it  will  be  found  in 
every  case  that  the  square  of  a  polynomial  is  expressed  by 
the  principle  employed  to  find  the  square  of  a  trinomial  ; 
namely. 

The  square  of  a  polynomial  is  equal  to  the  sum  of  the 
squares  of  its  terms  plus  twice  the  sum  of  the  products  of  all 
pairs  of  the  terms. 

Note.  A  systematic  way  of  naming  the  pairs  of  terms  in  a  poly- 
nomial is  as  follows :  Take  the  first  term  with  each  of  the  terms  that 


112  ELEMENTARY  ALGEBRA 

follows  it;  take  the  second  term  with  each  that  follows  it;  take  the 
third  term  with  each  term  that  follows  it ;  contirme  this  process  until 
next  to  the  last  term  is  taken  with  the  last. 

Thus,  the  sum  of  the  algebraic  products  of  all  pairs  of  terms  of 
the  polynomial  (a  +  b  -\-  c  —  d  +  e)  is 

ab  -\-  ac  -h  a(-  r/)  +  o.e  +  be  -\-  b(-  d)+  he  +  c(-  rl)  -\.  (^e  -\-  (-  d)e. 

EXEBCISE  42 

(Solve  as  many  as  possible  at  sight.) 
Square  the  following  polynomials  as  indicated : 
1.    (a  +  6  +  c?4-c?)2.  2.    {x-\-y  +  z-uy. 

3.    (m-\-n—p-\-  qy.  4.    (r  —  8  +  ^  -h  w)^. 

5.    (a  —  b  —  c-\-  dy.  6.    (m  —  n—p  —  q^. 

7.    {x-\-y^-z-\-iy,  8.    (mH-372-jo  + 2)2. 

9.    (a -h  5  +  <?  +  (f  +  e)2.  10.    (m—n—p  —  q  —  ty. 

95.  The  product  of    the  sum    and    difference    of    two 
numbers.     By  multiplication  we  find  that 
(a  +  &)(a-&)  =  a2_62. 

This  identity  may  be  expressed  in  words  as  follows  : 
The  product  of  the  sum  and  difference  of  two  numbers  is 
equal  to  the  square  of  the  first  minus  the  square  of  the  second. 

ILLUSTRATIVE  EXAMPLES 

1.  Find(a  +  l)(a-l). 

Solution.  (a  +  l)(a  -  1)=  a^  -  V  =  a^  -  1. 

2.  Find  (2  7w4-5y)(2m-5^/). 

Solution.       (2  m  +  5  y) (2  m  -  5  y)  =  (2  my  -  (5  y)^  =  4  m*  -  25  y^. 

3.  Find  (ax  -  6)  (ax  +  b) 

Solution.  (ax  -  b)(ax  +  6)  =  (ax^  -  (by  =  a^x^  -  b^. 

4.  Find  101  X  99. 

Solution.       101  X  99  =  (100  +  1)(100  -  1)  =  (100)2  _  i 

=  10000  -  1  =  9999. 


TYPE  PRODUCTS  AND  FACTORS  113 

5.    Find  (a  +  6  +  c)(a  +  ^  -  0- 


Solution,     (a  +  6  +  c)(a  +  6  -  c)  =  (a  +  6  +  c)(a  +  6  -  c) 

=  a2  +  2  a6  +  62  _  c\ 
6.    Find  (x  ■\-  y  —  z)(x  —  y  -\-  z). 


Solution,     (x  -\-  y  -  z)  {x  -  y  -\-  z)  =  {x  +  y  -  z){x  -  y  -  z) 

=  x^-{y-zy 
=  a;2  _  y2  _,.  2  y2  -  22. 

EXERCISE  43 

(Solve  as  many  as  possible  at  sight.) 
Multiply  as  indicated : 
1.    (x-{-V)(x-l).  2.    (a  +  2  6)(a-2  6). 

3.    (m-n)(m-\-n).  4.    (2 a  +  5  5)(2 a- 5  6). 

5.    (^ah-c){ah  +  c).  6.    (2  ah  +  c)(2ah  -  c). 

7.    (3a:^-2  2)(3a:z/H-2z).     8.    (4  a:- 7  z/)(4a:  +  7  ?/). 
9.    (_3a:+5^)(3a;4-5y).   10.    (- 1  +  a:)(rc  +  l). 
11.    ih-^yX\  +  ^y)-  12.    (a2  +  52)(a2-52). 

13.    (a2  +  62)(52_^2).  14.    (10a:?/z+3)(10a:«/3-3). 

15.  63  X  57.      Suggestion.    63  x  57  =  (60  +  3)  (60  -  3). 

16.  22  X  18.  17.    81  X  79. 
18.    37  X  43.                             19.    201  X  199. 

20.    202x198.  21.    (x^y^z)(x-\-y-z). 

22.    {a-h-{-c)(a-\-h-{-c).      23.    (- a -h  6  +  c)(a  +  64-c). 
24.    (a-\-h  — c){a  —  h  +  c). 

Suggestion,     (a  +  &  -  c)(a  -  6  +  c)  =  [a  +  (6  -  c)]  [a  -  (6  -  c)]. 

25.  (  — aH-6  +  c)(a— 5  +  <?). 

26.  (a-hl-^)(a-l  + J). 

27.  (2a  +  35-c?)(2a-36  +  (?). 

28.  (2a:-3  2/  +  2)(2a:  +  32/-2). 

29.  [a2  +  (5  +  c)2]  [a2  -  (6  +  O^J  • 


114  ELEMENTARY  ALGEBRA 

30.  (x3H-?/3)(2^-?/3). 

31.  (m*  —  n^)  (w*  +  w*). 

32.  (r6+l)(rS-l). 

33.  State  a  rule  for  telling  whether  or  not  a  binomial 
is  the  product  of  the  sum  and  difference  of  the  same  two 
numbers. 

34.  Which  of  the  following  may  be  expressed  as  the 
product  of  a  sum  and  difference  ? 

a4  _  ^  .  ^2  -  2  aft  +  52  +  1 ;     a^  +  f. 

35.  Find  in  two  different  ways  the  value  of 

(a  +  J)(a  —  5)  when  a  =  x+l  and  b  =  x—1. 

36.  Find  in  two  different  ways  the  value  of 

(x  -{-y^(x  —  y)-,  when  a;  =  2  a  +  3  and  y  =  '2a  —  b. 

37.  Find  in  two  different  ways  the  value  of 

{x  -\-  y^(x  —  y)^  when  x  =  a+  h-{-  c  and  y  z^a  +  b  —  c. 

96.  The  product  of  two  binomials  having  a  common 
term.     By  actual  multiplication  we  have  : 

X  -\-  a 
x-\-b 
x^+  ax 
-\-  bx  -\-  ab 


a;2  +  (a  +  b)x  +  ab 
Hence, 

(x+a)(jc  +  &)  =  x^  +  (a-h6)jc  +  fl&. 

This  identity  may  be  expressed  in  words  as  follows : 
The  product  of  two  binomials  having  a  common  term  is 
equal  to  the  square  of  the  common  term  plus  the  product  of 
the  algebraic  sum  of  the  unlike  terms  and  the  common  term^ 
plus  the  product  of  the  unlike  terms. 


TYPE  PRODUCTS  AND  FACTORS  115 

ILLUSTRATIVE  EXAMPLES 

1.  Find  the  product  oi  x  +  a  and  x  —  b. 

Solution,     (x  +  a)(x  —  b)=  x^-\-(a  -  h)x  +(«)(-  V) 
=  x2  +  (a  -  h)x  -  ah. 

2.  Find  the  product  oi  x—  a  and  x—b. 

Solution,     (x  -  a)(x  -b)=  x^  +(-  a-  b)x  +  (-  a)(-  b) 
=  x^  —(a  +  b)x  +  ab. 

EXERCISE  44 

(Solve  as  many  as  possible  at  sight.) 
Multiply  as  indicated  : 

1.    (ia  +  b)(a-\-c).  2.  (a  +  2)(a+l). 

3.    (2aH-l)(2aH-3).  4.  (3a  + 2)(3a -4). 

5.    (7H-2)(w  +  5).  6.  (a:  +  3)(2:H-2). 

7.    (a;H-2)(a;-3).  8.  (a;  4- 4) (a:  -  5) . 

9.    (a:-3)(a:-2).  10.  (x-5)(a;-7). 

11.  X^b-hc)(ab-\-d).  12.  (xi/  +  z^(xi/-\-l), 

13.    (2a5c  +  l)(2a6c+3).  14.  (i  a  +  1)(J  a  +  3). 

15.    (ia:+2)(Ja:-5).  16.  (3  a  -  l)(3a  -  2). 

17.    (a:  +  y  +  l)(2:  +  y  +  2).  18.  (a  +  6  +  2)(a  +  ^  +  3). 

97.  The  cube  of  a  binomial. 

By  actual  multiplication  we  have, 

a^-\-2ah  +  b^ 

a  -\-b 

a»  +  2  a%  +  ab^ 

+     a%  +  2ab^  +  b^ 
a^  -\- 3  a%  -\- 'S  ab^  +  b* 

Hence, 

(a+&)3  =  a3_j_ft3^3fl2ft  +  3afi2. 


116  ELEMENTARY  ALGEBRA 

By  the  use  of  this  identity  the  cube  of  any  binomial  can 
at  once  be  written  as  in  the  following : 

ILLUSTRATIVE  EXAMPLES 

1.  Find  the  cube  of  a  —  h. 

Solution,     (a  -  &)8  =  a»  +  (  -  ft)8  +  3  a2(  -  6)  +  3  a(  -  6)2 
=  a8-&8-3a26  +  3a62. 

2.  Find  the  cube  of  2  a  +  3  6. 

Solution.     (2  a  +  3  &)8  =  (2  ay  +  (3  &)»+  3(2  a)2(3  6)  +  3(2  a)(3  hy 
=  8a8  +  27  68  +  36a26  +  ^ah\ 

3.  Cube  98  by  the  use  of  the  identity  of  section  97. 
Solution.     (98)8  =  (100  -  2)8 

=  (100)8  -f(-  2)8  +  3(100)2(-  2)4-  3(100)(-2)« 
=  1000000-8-60000  +  1200 
=  941192. 

EXERCISE  45 

(Solve  as  many  as  possible  at  sight.) 

Cube  the  following,  as  indicated  : 

1.    (x  +  yy.  2.    (x-yy,  3.    (a +  1)8. 

4.    (a -1)8.  5.    (re -2)8.  6.    (a: +  2)3. 

7.    (2  a +  5)3.  8.    (2a;  +  3y)3.  9.    (3-2ic)8. 

10.    (4iir-33/)3.      11.    (99)3.  12.    (101)8. 

13.    Is  8  a3  + 12  a%  +  6  a62  +  63  a  perfect  cube  ? 

EXERCISE  48.    REVIEW 

(Solve  as  many  as  possible  at  sight.) 

By  use  of  type  identities  perform  the  indicated  multi- 
plications in  examples  1-33. 

1.    a(2-3  6).  2.    2  6(3a;  +  52/).     3.    (x+\y. 

4.    (a; -7)2.  5.    (5  a; +2)2.  6.    (3  a; -4)2. 

7.    (6a;  +  32)2.  8.    (2w-7w)2. 


10. 

(^_9)(a:-l). 

12. 

(a:-ll)(a:+3). 

14. 

(2  2:-ll)(2a;  +  ll). 

16. 

(x-hl)(x-\-a}. 

18. 

(2r  +  w)(2r-7i). 

20. 

(2r  +  l)3. 

22. 

{S-^a-by. 

24. 

(xy-lX^^-^^y 

26. 

(2-32/)3. 

28. 

(l-h2:-^^)(H-2:  +  y). 

TYPE  PRODUCTS  AND  FACTORS  117 

9.    (a:  +  5)(2:4-3). 
11.    (a:4-10)(2:-2). 
13.    (a:-9)(rc+9). 

15.      (4/71  —  7l)(4  7W  +  W). 

17.  (a:- 6)(a;H-4). 

19.  (2  2:+?/ +  5)2. 

21.  (l  +  3c)(l  +  4(?). 

23.  (a:-y  +  l)(a:-y-|-2). 

25.  (a: +  3)3. 

27.  (2a— 5c)(2a  +  5<?). 

29.  (xy  —  yz  +  zx)(xy -\- yz  —  zx). 

30.  (l  +  a  +  2a2)2.  31.    («+3)(^_i). 

32.    (a  +  f)(a-2).  33.    (ia  +  l5)(Ja-^6). 

34.  Verify  the  identity : 

(a2  +  62)(2:2  +  y2)  ^  (^^  ^  5^)2^  (^y  __  5^)2. 

35.  Verify  the  identity  : 

(a2  ^  52  4.  c2)(^2^  y2^  22)  =  (aa:  +  %  +  C2)2+  (hz  -  cy^ 

+  (ca;  —  azy  +  (ay  —  hxy. 

Simplify   the   following  by   performing   the   indicated 
operations  and  when  possible  uniting  terms : 

36.  (l  +  a;)2-(a;-l)(a:  +  l). 

37.  (a  +  5)2(a  -  6)2. 

Suggestion,     (a  +  6)2(a  -  hy  =[(a  +  h){a  -  6)]2 

38.  (2x+3«/)2(2a:-3  2/)2. 

39.  (rr2  4.a.4.i)(2^2-ir  +  l). 

40.  (a^^x-k-lX^-x^-l^ia^-x^^V). 

41.  (a  +  5  +  c)(a  +  6- c)(a-  5  +  <?)(- a  +  6  +  <?). 
Suggestion,     (a -\- b  +  c)(a  -f  6  -  c)(a  -  b  +  c)(-  a  +  b  -\-  c) 

=  [(a  +  6  +  c)(a  +  6  -  c)][(a  -  6  +  c)(-  a  +  6  +  c)]. 


118  ELEMENTARY  ALGEBRA 

42.  (a-^h  + e  +  d)(a-^h  +  c- d). 

43.  a(b  — c) +  b(^c  — a)+ c(a  —  h). 

44.  2a;(3y-42)+3?/(42-2a:)  +  4  2(2a;-3y). 

45.  (a  -f  ^  +  <?)(«^  +  ^^  +  ^'^  -  5(?  -  <?a  -  aft). 

46.  (a2  -h  a6  -h  ^2)(a  -  b').        47.    (a2  -  aft  +  b^)(a  +  ft). 

Factors 

98.  Integral  algebraic  factors.  In  this  chapter  we 
shall  consider  only  the  integral  factors  of  integral  expres- 
sions of  the  more  common  form. 

99.  Factors  of  integral  expressions.  By  factors  of  an 
integral  expression  are  meant  those  integral  expressions 
which  when  multiplied  together  will  produce  the  given 
expression. 

Note  1.  The  word  integral  refers  only  to  the  literal  part  of  the 
expression.     Thus,  |  a  and  ^  are  integral  expressions. 

Note  2.  A  factor  of  each  of  two  or  more  expressions  is  called  a 
common  factor  of  the  expressions. 

100.  Prime  number  and  prime  expression.  In  arith- 
metic a  prime  number  is  defined  as  an  integer  which  is 
divisible  by  itself  and  1  and  by  no  other  integer ;  as,  3, 5,  7. 
In  elementary  algebra  an  integral  expression  with  integral 
coefficients  is  said  to  be  prime  when  it  is  divisible  by  itself 
and  1,  and  by  no  other  integral  expression  or  integer. 

Thus,  (x^  +  y^)  is  a  prime  expression. 

Note.  An  integer  or  an  integral  expression  with  integral  coeffi- 
cients may  often  be  expressed  in  more  than  one  way  as  a  product  of 
factors,  but  it  can  be  expressed  in  only  one  way  as  the  product  of  its 
prime  factors. 

For  example,  24  =  12x2  =  8x3  =  2x2x2x3. 

When  an  expression  is  to  be  factored  it  is  understood  that,  in 
general,  its  prime  factors  are  required. 


TYPE  PRODUCTS  AND  FACTORS  119 

101.  Use  of  type  forms.  The  type  forms  in  multiplica- 
tion are  of  great  service  in  giving  the  factors  of  an  inte- 
gral expression.  In  many  cases  they  furnish  the  clue  as 
to  the  kind  of  factors  to  expect. 

102.  Factors  of  monomials.  The  literal  factors  of  a 
monomial  can  always  be  seen  at  a  glance. 

Thus,  the  factors  of  —  2  a%'^c  are  —  2,  a,  a,  a,  b,  b,  and  c. 
Note.     Here,  as  elsewhere,  the  sign  before  the  monomial  is  to  be 
regarded  as  belonging  to  the  numerical  coefficient. 
Thus,  -a%=(-l)a^b. 

103.  The  converse  of  the  distributive  law.  We  learned, 
section  60,  that 

w(a  -{•  b  -{- € -\-  d)  =  ma -\- mb  -\- mc -\-  md ; 
hence,  conversely, 

ma -\- mb -{- mc -\- md  =  m(a  +  6  +  c  -h  (f). 

From  this  identity  it  follows  that: 

A  monomial  which  is  a  factor  of  every  term  of  an  expres- 
sion is  a  factor  of  the  whole  expression. 

Note.  The  first  step  in  factoring  an  integral  expression  is  to 
remove  all  monomial  factors. 

ILLUSTRATIVE  EXAMPLES 

1.  Factor  ^x  +  ^y. 

Solution.  3  2;  +  6  2/  =  3(a:  +  2  y). 

2.  Factor  mx  -\-  my  —  m. 

Solution.  mx  +  my  —  m  =  m(x  -f  y  —  1). 

3.  Factor  a(l  —  m)  4-  a(m  -\-  n). 

Solution.     a(l  -  w)  +  a(m  +  n)  =  a[(l  -  m)  +  (m  +  n)] 

=  a[/  -  m  +  m  -\-  n^=  a{l  -\-  n). 

4.  Factor  (a  +  5)  (2:  +  ?/)  -  (a  +  5)  {y+z). 

Solution.     {a^b){x  +  y)-{a  +  b){y  +  z)  =  (a-{-b)l{x^y)-{y  +  z)-\ 

=  (a  +  b)\_x  +  y  -  y  -  z'] 
=  ia-\-h){x-z). 


120  ELEMENTARY  ALGEBRA 

5.    Factor  (a  +  i)(6-c)  +  (a-|-6)(<?-a)H-(a-f-J)(a--6). 
Solution,     (a  -\-  b)(b  -  c)-\-(a -\-  b)(c  -  a)  +  (a  +  6)  (a  -  b) 

=  (a  +  b)[b  -c-^c-a  +  a-b]. 

=  (a  +  b)0  =  0. 

EXERCISE  47 

(Solve  as  many  as  possible  at  sight.) 
Factor  the  following  expressions  by  removing  the  mo- 
nomial factors : 

1.  4iX-\-12y.  2.    am  —  an. 

3.  5  a;  4- 10  ^  —  15  2.  ^.  px  —  py  -\-  pz. 

5.  ax  -{■  ay  -\-  az,  6.    |  a  4- 1  J  —  f  c. 

1.  2  ax— 2  ay.  8.    3  aw  —  6  aw  4-  9  ar, 

9.  12x-\-^0xy.  10.    x^y  —  xy'^. 

11.  \ab+^ac,  12.    -3a^*H-12a:. 

13.  ax  +  a.  14.    ax—  ay  -\-  a. 

15.  4ca;  +  6cy-2c.  16.    a% -2  a^b^ +  ?>  ah^. 

17.  a:2y25  +  2  aj?/%  —  3  a:^^^^  is.    —  abc  —hc  —  h. 

19.  5  a^  +  10.  20.    a:?/  —  x. 

21.  xy  ■\-  xz  —  X.  22.    a(ft  +  <?)  4-  «(aJ  4-  y)» 

23.  (^x  +  y)(J>^c)  +  (iy^z^{h-{-c), 

24.  (a  4-  h)x  —  (a  4-  ^)«/. 

25.  (a4-^>-(a4-i)«/H-(a4- J>. 

26.  (a4-^)<?4-(«4-*)c2-(a4-^)e.     • 

27.  (a;4-y)(2a+ft)4-(a^4-?/)(^»-a). 

28.  (3  a  +  2  5)a  4-  (3  a  4-  2  J)5  -  (3  a  4-  2  6)(a  -  6). 

29.  axyz  — ayux -\- axut. 

30.  2(a  4- ^)<?  4- 2(a  4- ^)c?. 

31.  2{a-h)cd-2{a  +  h)ia-1)), 

32.  2(a:  +  ^)«  -  4(a:  4- ,y)e. 


TYPE  PRODUCTS  AND  FACTORS  121 

33.  SQa  +  h)c+6(^a-^b')d, 

34.  2(3  a  -  2  a:)(3  «  +  2  or)  +  3(3  a  -  2  2;)(2  2:  -  a). 

35.  a(6  +  l)+<6+l). 

36.  x(2/-l)  +  ?/(y-l) +  («/-!). 

37.  a;(2-l)4-«/(2 -l)  +  2- 1. 

38.  x(a—l)-\-i/{a-l^-a  +  l. 

The  expression  may  be  written  x(a  —  1)  +  y(a  —  1)  —  (a  —  1). 

39.  a(^x  —  y^  —  x-\-  y. 

40.  a(^y  —l)-\-by  —  h. 

41.  a(h  +  c)-\-2b-\-2c. 

42.  2(x  —  y)-\-^x  —  4:y. 

43.  3  a(?/  -\-z)-^hy  —  Zhz. 

44.  (a  +  5)(a;  +  y^  -  (a  +  5)2(a;  +  ^)-f-(a  +  6)(a;  +  y). 

45.  (a  +  hy  4-  (a  +  J)c. 

46.  2(a  +  6)^2  -  2(a  +  ^)26-. 

47.  (w+w)Cjt?  +  ^)24-(m  +  w)(p  +  9)  — (wH-w)2(jo-f-5'). 

104.  Factors  found  by  grouping  terms.  Examples  37 
to  43  inclusive  of  exercise  47  furnish  simple  instances  of 
the  grouping  of  terms.  In  each  of  these  examples  the 
grouping  required  merely  the  insertion  of  a  single  set  of 
parentheses.  In  general,  in  factoring  an  integral  expres- 
sion of  four  or  more  terms  by  the  aid  of  rearrangement 
and  grouping  of  terms,  those  terms  should  be  grouped 
together  which  have  a  common  monomial  factor.  Two  or 
more  ways  of  grouping  may  be  possible,  and  some  of  these 
ways  may  lead  to  a  common  factor  and  others  may  not. 
It  is  not  possible,  however,  to  give  any  simple  rule  for 
proper  grouping  when  several  ways  of  grouping  exist,  but 
a  careful  study  of  the  following  illustrative  examples  will 
prove  helpful. 


122  ELEMENTARY  ALGEBRA 

ILLUSTRATIVE  EXAMPLES 

1.  Factor  ax -^  bx  -\- a^  -\-  hy. 

Solution,     ax  -\- hx  -{■  ay  -\- hy  =  (ax  +  hx)  +  (ay  +  hy) 

=  x(a -ir  b)  Vy(a  +  h) 
=  (a-\-h)(x-{-y). 

2.  Factor  ax—hx  —  ay  —  h-\-a-\-hy. 

Solution.     ax  —  hx  —  ay  —  h-\-a-\-hy  =  {ax  —  ay  +  a)-\-{  —  hx  —  h-\-hy). 

—  a{x  -  y  -\-  1)—  h{x  -  y  -{■  V) 
=  {x-y  +  l){a-h). 
Another  Solution. 

ax  —  hx  —  ay  —  h  -{■  a  -^  hy  =  (ax  —  hx)  +  (—  ay  -\-  hy) -\- (—  h  -^  a) 
=  x(a  —  h)—  y(a  —  h)-\-(a  —  h) 
=  (a-h)(x-y-^l). 

3.  Factor  a^ -\- a%  +  ah^ -\- b^. 

Solution.     a8  +  a%  +  ah^  +  h»  =  a^a  +  h)+  h\a  +  h) 


EXERCISE  48 

Factor : 

1.  2a-\-2b  -\-  ac  +  hc.  2.  ax-^  x  +  a-\'l, 

3.  am  -^  an  —  4:  m  —  4:71,  4.  by  -{-  y  —  2b  —  2, 

5,  Ip  +  qm  -\-  mp  H-  ql.  6.  am  —  cm -{-  cu  —  au, 

7.  xy-\-2y-2x-4.  8.  6  -  9  a  +  46  -  6a6. 

9.  6  m  -  3  -  2  am  +  «.  10.  6  rs  -  9  /•  -  2  «2  4.  3  «. 

11.  ^x^+\bxy-\-QyK 

Suggestion.     9  a:^  +  15  2:^  +  6  ^2  =  3(3  x^  +  5  ary  +  2  y^) 

=  3(3 x^  +  ^xy  -^2xy  +  2y^) 
=  3[(3x2  +  3x3^)  +  (2x3/-l-2y2)] 

=  ^lZx(x  -\-  y)-\-2y(x  +  y)-\. 

12.  a^  -\-  ah  -\-  a  —  ca  —  cb  —  e  +  a  -\-  b  +  \, 

13.  2^2  — a:^ -I- a;— rrz  +  22/ —  z  4-a:  — y -h  1. 


TYPE  PRODUCTS  AND  FACTORS  123 

14.    a{x-y^-^h{y  -x).  15.    2(a  -  h)-  x(h  -  a), 

16.    a^-x'^  +  x-l.  17.    T^-\-x^+x-\-\. 

18.  82:3_4^2^2a:-l. 

Suggestion.     8  ar^  -  4  a;^  +  2  a;  -  1  =  (2  a:)8  -  (2  xy+  (2  x)  -  1. 

19.  %a^  +  ^x^+2x+\. 

20.  aa;^  —  ha^  -\-  ax  —  cx^  —  bx—  ex. 

21.  a:2-3  3:  +  2. 

Suggestion.     x^-Sx  +  2  =  x^-x-2x  +  2. 

22.  flw:  +  a^  +  ^2  4-  ^a:  +  ^^  -I-  ^2  H-  ca:  4-  CI/  +  cz. 

23.  a^  +  a52  -|-  a%  -\-  b^.  24.    x^  4-  a:?/^  —  a:^^  —  y^. 
25.    a^^;  _  2  —  ^2  _|_  2  a:.            26.    15  —  6  a:  —  10  ^  +  4  xy. 
27.    1  —  abed  -\-  ae  —  bd.  28.    ??i2n2-f  m2p2-)_^7^2_|_^2^2 
29.    (2:2  _  ^2^  2  —  (^2  _  ^2)  2-^ 

Suggestion.     First  remove   the   given   parentheses,  then   group 
terms. 

20.    \  —  abx^  —  (^a  —  b)x,        31.    x^  •\-{a-{-b')x-\- ab. 

32.    (ia^^-y^)c+(b^+(P'')a.      33.    aijy^^- (^)-b{<^+ a^). 

105.   Trinomial  squares.     We  learned,  §  92,  that 

(a  +  5)2=  ^2  ^  2  a5  +  2>2  and  (a  -  6)2=  «2  _  2  a5  +  ^2  ; 

conversely 

(1)  fl2  +  2a&  +  62  =  (a+6)2. 

(2)  fl2-2a&+2r^  =  (a-6)2. 

From  identities  (1)  and  (2)  it  is  obvious  that 

If  two  terms  of  a  trinomial  are  perfect  squares  and  the 

third  term  is  equal  to  plus  or  minus  twice  the  product  of  the 

square  roots  of  the  other  two  terms,  then  the   trinomial   is 

the  square  of  a  binomial. 

When  two  terms  of  a  trinomial  are  perfect  squares  and 

the  third  term  is  equal  to  plus  or  minus  twice  the  product 


124  ELEMENTARY  ALGEBRA 

of  the  square  roots  of  the  other  two  terms,  the  binomial 
square  root  —  that  is,  the  binonlial  which,  when  squared 
is  equal  to  the  given  trinomial  —  may  be  found  by  taking 
the  positive  square  roots  of  the  two  terms  of  the  trinomial 
which  are  perfect  squares,  and  adding  one  of  these  square 
roots  to  the  other,  or  subtracting  it  from  the  other,  accord- 
ing as  the  third  term  of  the  trinomial  is  positive  or 
negative. 

Remark,  (a  —  by=(b  —  a)^  since  a  —  b  and  b  —  a  differ  only  in 
sign,  and  (+  a)2=(—  0)2  whatever  expression  may  be  represented  by 
a.  However,  it  is  customary  to  write  a^  —  2ab  -\-  b^  equal  to  either 
(a  ~  by  or  (b  —  ay  at  pleasure  and  not  to  write  it  equal  to 
[±(a  -  6)]2.  Similarly,  a^  -\-2ab  +  b^  is  written  (a  +  by  and  not 
[±(a  +  6)p. 

ILLUSTRATIVE   EXAMPLES 

1.  Factor  a2+ 2a +  1. 

Solution.  a2  +  2  rt  +  1  =  (a)2+  2(a)  (1)  -f  (1)2,  which  satisfies  the 
conditions  for  a  perfect  square. 

,'.a^-\-2a  +  l=(a  +  iy.  [§105,1] 

2.  Factor  x^  —  4:  x^ -\- 4. 

Solution,  x*  -  4  x2  4-  4  =  (x^y  -  2(2)  (x^)  +  (2)2,  which  satisfies 
the  conditions  for  a  perfect  square. 

.-.  a:*  -  4  a;2  +  4  =  (^2  -  2)2.  [§  105,  2] 

Remark,     (x^  -  2)2  may  be  written  (2  -  x2)2.        [§  105,  Remark] 

3.  Factor  4  r*- 12  2:2^  + 9/. 

Solution.     4:x^  -  12xy  +  9y^  =(2xy-  2(2x) (3 y)  +  (3 yy 

=  (2x-dyy. 

4.  Factor  (a  -f  6)2-  6  (a  +  5)(c  -  c?)  +  9((7  -  dy. 
Solution,     (a  +  6)2-  6(a  +  b)(c  -  rf)+  9(c  -  dy 

=  (a  +  by  -  2(a  +  6)[3(c.  -  d>]  +  [3(c  -  d)Y 
=  [(a  +  &)-3(c-rf)]2 
=  (a  +  6_3c  +  3rf)2 


TYPE  PRODUCTS  AND  FACTORS  125 

5.    Factor  x^  —  x -\-  \. 

Solution.     x^-x  +  l  =  x^-  2(i)  x  +  Qy. 

Remark.  Although  not  all  of  the  numerical  coefficients  of 
x^  —  X  -\-  i  are  integral,  yet  the  terms  of  the  trinomial  satisfy  the 
conditions  for  a  perfect  square. 

EXERCISE  49 

(Solve  as  many  as  possible  at  sight.) 
Factor  the  following  trinomials : 

1.  m2  -f-  2  mw  4-  n^.  2.  r^  —  2  rs  -f-  «2. 

3.  aP'j-2x-{-l.  4.  a2-h4a  +  4. 

5.  7w2-6m  +  9.  6.  4:a^-\-4:x-{-l. 

7.  x^-6xi/-\-9y^,  8.  x^  +  x-\-\. 

9.  a2-a+i.  10.  9^:2+32:  +  ^ 

11.  a^+2rc2  +  l.  12.  4a*-h20a2+25. 

13.  25a4-20a2  4.4.  14.  x^i/^ -\- 2x1/ -{■  1^ 

15.  4a252-4a5  +  l.  '     16.  x^ -\- 2xf -\- 1/^. 

17.  a254_2a52-|_l.  is.  2^/ -  2  a;?/V  +  2*. 

19.  x^-Qxf-\-9f.  20.  4  a2  _  20  a52  +  25  5*. 

21.  9  x^t/^z^ -\- 6  axi/z -{■  a^.         22.  a2  -  12  a -f- 36. 

23.  25m24-70ww+49n2.  24.  (p^qy+2(p  +  q}-\-l. 

25.  a2«  +  2  a"»5"  +  ^". 

Suggestion,     a^^  +  2  a'^b''  +  J^n  _  (am)2_^  2(a'")  (&»*)  +  (b'^y. 

26.  a:2p  —  2  a:^?/'  +  ^2«.  27.    a^  +  4  aH""  +  4  62n. 
28.    4  a63  -  4  ^252  _^  ^5,  29.    7?y  -2  :^y  +  a;y. 

30.    a25^  +  4  aj^  +  4  5^2.  31.    2:2^22  _  14^^22 +  49^2, 

32.    o?-V\%^-\-^\x.  33.    w2wjt?4-20mwp  +  100rip. 

34.  (a:+^)2_6(2:  +  y)(a+i)+9(a+5)2. 

35.  2:2_iOx(?/  +  2)+25(?/  +  z)2. 


126  ELEMENTARY  ALGEBRA 

36.  a^-^b^+c^  +  2hc-\-2ca-\-2 ah. 
Suggestion,     a^  +  b^  +  c^  +  2bc  +  2 ca  -\-  2 ab 

=  a2  +  2(b  +  c)a  +  (62  +  26c  +  c«). 

37.  a^  +  h^-^(^-\.2hc^2ca-2ah, 

38.  4a2-28a(6  +  c)4-49(5+c)2. 

39.  49a:V+28a:«/2(3^4.2)+4i/2(y  +  2)2. 

40.  l-6(a-5)+9(a-6)2. 

106.  The  difference  of  two  squares.  We  learned, 
section  95,  that 

(a  +  6)(a  —  5)  =  a2  _  52 .    hence,  conversely, 
a2-63  =  (a4.6)(a_5). 

From  this  identity  we  infer  the  following  rule  for 
factoring  the  difference  of  two  squares : 

Rule.  Find  the  positive  square  root  of  each  of  the  two 
squares  and  form  the  sum  and  the  difference  of  these  square 
roots  in  the  order  in  which  their  squares  occur  in  the  expres- 
sion. The  sum  and  the  difference  of  the  square  roots  are  the 
two  factors, 

ILLUSTRATIVE  EXAMPLES 

1.  Factor  9a^-25. 
Solution.     9  x2  -  25  =  (3  x)^  -  (5)2. 

The  positive  square  roots  of  the  squares  are  3  x  and  5.  The  sum 
of  the  square  roots  is  3  x  +  5  and  the  difference  is  3  a:  —  5. 

.-.  9ar2  -  25  =(3x  4- 5)(3ar  -  5). 

2.  Factor  a2  -  (5  -  c)2. 

Solution.  The  positive  square  roots  of  the  squares  are  a  and  b  —  c. 
The  sum  of  the  square  roots  is  a  +  (b  —  c),  and  the  difference  is 
a  —  (6  —  c)  ;     that  is,  a  +  6  —  c  and  a  —  b  +  c. 

.-.  a2  -(6  -  c')2  =(a  +  b-  c)(a  -  6  +  c). 


TYPE  PRODUCTS  AND  FACTORS  127 

In  practice,  the  work  of  factoring  a^  —(b  —  cy  may  be  arranged 
thus: 

a^-(b-  c)2  =  [a  +(b  -  c)]  [a-(b-  c)] 
=  (a  -{-  b  —  c)(a  —  b  +  c). 

3.  Factor  a2  _  1  52. 
Solution.  a2  _  A  52  =  ^2  _  (2  5)2 

4.  Factor  a*  —  b^. 

Solution.  •  a*  -  6*  =  (a^y  -  (b^y 

=  (a^  +  b^)(a^-b^) 

=  la^  +  b^X^  +  b)(a  -  b). 

EXERCISE   50 

Factor  : 

1.  m^  —  n^.  2.    2^2  _  y2  Z,    W  —  r^. 

4.  a^-\.  5.    1-32.  ,       6.    m2-4. 

7.  9-^2.  8.    42:2_^2,  9.    ^2_1622. 

10.  2^2  _  100.        11.    252;2-64?/2.       12.    ^m^-\^n'^, 

13.  w*-w2.         14.    a;6_^2,  15.    36a:4-49«/8. 

16.  25  a254  _  36  52^4,  17,  ^  -  x. 

18.  a^-4ir2,  19.  2^-?/*. 

20.  ^  —  {y-\-  25)2.  21.  ( w  H-  w)2  -  jt?2, 

22.  {a  -f  5)2  -  9.  23.  2^2  -  I  y2. 

24.  (a+6)2-((?+(^)2.         25.  l_(a:-y)2. 

26.  4(a-6)2-9(a  +  5)2.   27.  25(2: -^)2- 36(2^4-^)2. 

28.  (&-Q>-cy.  29.  9c2_(a-}-6  +  (?)2. 

30.     a2+2a(?  +  c2-52.  31.     a;2_l_2^_^2. 

32.    2^2  ^10  a;  4.  25- 25  «/2.   33.    \-m^ -"Imn-n^. 

107.   Trinomials  of  the  form  x2  4-  ex  +  {/.      From   sec- 
tion 96  we  have, 

{x  4-  a)(2;  +  5)  =  2;2  +  (a  +  h^x  +  ah  ;  conversely, 
Jt2  4- (a  4- &)jir  +  a6  =  (jr  +  a)(jr  +  6). 


128  ELEMENTARY  ALGEBRA 

From  this  identity  we  see  that : 

Any  trinomial  of  the  form  a^-\-  cx-{-  d  can  be  factored 
when  c,  the  coefficient  of  x,  is  the  sum  of  two  expressions, 
and  d,  the  last  term,  is  the  product  of  the  same  two  expressions. 

Remark.  When  c  and  d  are  given  integers  and  not  too  large,  it  is 
possible  to  determine  by  inspection  whether  two  other  integers  a  and  h 
exist  such  that  a  +  b  =  c  and  ab  =  d.  When  t^o  such  integers  are 
found  the  factors  oi  x^  -\-  ex  -\-  d  are  {x  +  a)  and  (x  +  b). 


ILLUSTRATIVE  EXAMPLES 

1.  Factor  x^ -{- 1  x -\- 12. 

Solution.     The  two  integers  whose  sum  is  +  7  and  whose  product 
is  +  12  are  evidently  -f  3  and  +  4. 

.      ...  a:2-<-7a;  +  12=(x  +  3)(x  +  4).  [§107] 

2.  Factor  q^-{-x  —  12. 

Solution.     The  two  integers  whose  sum  is  +  1  and  whose  product 
is  —  12  are  evidently  +  4  and  —  3. 

.-.  a:2  +  a:  -  12  =  (x  -  3)(a:  +  4).  [§  107] 

3.  Factor  a;^  -  9  ic  +  20. 

Solution.     The  two  integers  whose  sum  is  —  9  and  whose  product 
is  +  20  are  evidently  —  4  and  —  5. 

.-.  a;2  -  9  a:  +  20  =  (x  -  4)(a:  -  5).  [§  107] 

4.  Factor  x^  -\-  a(h  —  c)x  —  c^hc. 

Solution.     The  two  expressions  whose  sum  is  ab  —  ac  and  whose 
product  is  —  aV)c  are  evidently  ab  and  —  ac. 

.'.  x^  +  a(b  -  c)x  -  a%c  =  (a:  +  ab){x  -  ac).  [§  107] 

5.  Factor  x^-2  x^y^  -  15  y^. 

Solution.     The  two  monomials  whose  sum  is  -  2  y*  and  whose 
product  is  —  15  y^  are  —  5  y^  and  3  y^. 

...  x^-2  xhf  -\by^=  {x^  -  5  y'^ix^  +  3  ^2).        [§  107] 


TYPE  PRODUCTS  AND  FACTORS  129 

BXBROISB  51 

Factor  the  following : 

1.    a2  +  3a  +  2.  2.  y2  ^  2  ^  -  3. 

3.    22_2_6.  ^.  h^-Qh  +  b. 

5.  jo2  +  6jD  +  8.  6.  ^-7c?  +  12. 

7.    x^-bx-1^.  8.  «2_4^_45. 

9.    52-12  5  +  32.  10.  22  +  13^  +  30. 

11.    m2  +  4m-221.  12.  .c2  +  18  a:  4- 72. 

13.   jp2_i0jp_ll.  14.  2^ -10  a; +  24. 

15.    ;r2-27a;  +  50.  16.  a2-18a  +  80. 

17.    x^-l^x-\-^\.  18.  «24.8a_209. 

19.    a2+9a-36.  20.  2^2 -2; -2. 

21.    62  +  27  5  +  152.  22.  w2-16m  +  55. 

23.    22.^52-24.  24.  2:2  ^_  10  a:  -  39. 

25.    a2_i2a_l33.  26.  w2- 28  m +  171. 

27.    2:2  _  78  2; +  365.  28.  2^2  _^  10  a:  -  119. 

29.    2:2  +  3  2; -154.  30.  22_|.62-91. 

31.    a^  +  a-mO.  32.  Z2_j.24Z+23. 

33.    2:2  _  26  a;  -  155.  34.  a2_i8^_i9. 

35.    a^y^  +  4  2:2^/2  _  5  xy.  36.  x^-{-(a  +  S)x  +  3  a. 

37.    a2_(2- 5)a- 2  5.  38.  «/2 -(5  +  ^)^  +  5  2. 

39.    ^2  _j_  2;(?/  _  ;2)^  _  x^yz,  40.  2:2  +  3  2:  —  a2:  —  3  a. 

41.    2:2  _  ^2;  +  3  ^^  _  3  ^^  42.  a^  _|_  2  52:  —  C2:  —  2  5(7. 

43.     6/2  4-  3  a5  +  2  52.  44.  2:2  _  6  ^^2  +  5  ?/*. 

45.    a*  +  4  «2J  -  221  52.  46.  X^-{-S  2:2/  _^  15  2^. 

47.  2:2- 2a2:-2  52:  +  a2_|_2a5. 

48.  2:2^^2:— 52:  +  6  —  3  a. 


130  ELEMENTARY  ALGEBRA 

108.  The  general  quadratic  trinomial  ax^  -\-  bx  -h  c.     By 

actual  multiplication  we  have 

(^px  -f  q)  (rx  +  «)  =  pro^  +  (ps  -f  qr^x  +  qs; 
conversely, 

prx^  -h  (ps  +  qr}x  -^qs  =  (px  +  q)(rx  +  s). 

We  observe  in  the  first  member  of  this  identity  that 
the  coefficient  of  x  is  the  sum  of  two  terms  +  ps  and  -h  qr 
whose  product  is  -i-pqrs  and  that  the  product  of  the  coeffi- 
cient of  x^,  namely  pr^  and  the  last  term,  namely  qs,  is  also 
+pqr8.  These  facts  furnish  a  clew  which  is  of  assistance 
in  factoring  a  trinomial  of  the  form  aa^  +bx'^c  whenever 
it  is  possible  to  express  b  as  the  sum  of  two  numbers  whose 
product  is  equal  to  ac. 

Note.     In  the  foregoing,  we  have, 

a  =  pr,   h  =  ps  ■{■  qr,   c  =  qs. 

Therefore,  V^  =  ph^  +  2  pqrs  +  qh-^  ]  ^ 

and,  4  ac  =  ^pqrs ; 

hence,  h^  —  ^ac  =  p^s^  —  2pqrs  +  q^r^  =  (ps  —  qry. 

If,  therefore,  in  aaj-^  -i-  bx  -\-  c,  the  square  of  the  coefl&cient  of  x 
minus  four  times  the  product  of  the  coeflBcient  of  x^  by  the  last  term 
is  not  a  perfect  square,  it  is  not  possible  to  express  the  quadratic 
ax^  +  bx  -\-  c  a,s  the  product  of  two  rational  factors. 

The  converse  of  this  statement  (namely,  that  when  b^  —  4:  ac  is  a 
perfect  square,  integral  values  of  p,  q,  r,  and  s  exist  such  that  a  =  />r, 
^  =z  ps  -\-  qr,  c  =  qs)  will  be  proved  in  a  later  chapter.  (The  letters  a, 
b,  c  are  assumed  here  to  represent  positive  or  negative  integers,  or 
integral  expressions.) 

ILLUSTRATIVE  EXAMPLES 

1.    Factor  6  2^2  .,.19 ^^10, 

Solution.  If  possible,  we  must  find  two  integers  whose  sum  is  19 
and  whose  product  is  6  x  10,  or  60.  These  integers  are  evidently  15 
and  4. 


TYPE  PRODUCTS  AND  FACTORS  131 

Therefore,  6  a:^  +  19  x  +  10  =  6  a:^  +  ISx  +  4  x  +  10 

=  S  x(2  X  +  5)  +  2(2  X -{■  5) 
=  (2x+  5)(3a;  +  2). 

That  is,      6x2  +  19a:  +  10  =(2x  +  5)(3x  +  2). 

2.  Factor  10  2:2  _  7  a- _  12. 

Solution.     If  possible,  we  must  find  two  integers  whose  sum  is 

—  7  and  whose  product  is  10(—  12),  or  —  120.  Since  their  product 
is  negative,  one  of  these  integers  is  positive  and  the  other  negative. 
Moreover,  the  larger  integer  is  negative,  since  the  sum  of  the  two 
integers  is  negative,  namely  —7.     The  required  integers  are  evidently 

-  15  and  +  8,  since  -  15  +  8  =  -  7,  and  (-  15)(8)  =  -  120. 

.-.  10x2  -  7x  -  12  =  10x2  _  I5x  +  8x  -  12 
=  5x(2x-3)+4(2x-3) 
=  (2x-3)(5x  +  4). 
That  is,  10x2  -7x-  12  =(2x  -  3)(5x  +  4). 

(In  this  example  a  =  10,  6  =  -  7,  and  c  =  - 12.  ^>2  _  4  q^.  =  629  =  23*. 
Since  b^  —  4:ac  is  a  perfect  square,  the  given  expression  can  be  ex- 
pressed as  the  product  of  two  rational  factors.) 

3.  Factor  acx^  +  (^bc  —  a^x  —  b. 

Solution.  If  possible,  we  must  find  two  expressions  whose  sum 
ia  be  —  a  and  whose  product  is  —  abc.  The  required  expressions  are 
evidently  be  and  —  a.     Hence, 

acx2  +  (be  —  a)x  —  b  =  aex^  +  bex  —  ax  —  b 
=  ex(ax  -\-  b)  —  (ax  +  b) 
=  (ax+ 6)(cx- 1). 
Note.     See  problem  29,  exercise  48. 

EXEBCISB  52 


Factor : 

1. 

Qa^-x-1. 

3. 

6x^-\-x-5. 

5. 

102:2- 13a: -3. 

7. 

3p^-7p-Q, 

9. 

622  +  113  +  3. 

11. 

262+116-21. 

2. 

Ux^^^x-1, 

4. 

14arJ  +  2:-3. 

6. 

9a2_9^_4. 

8. 

6m2-5wi-4. 

10. 

862_i45_i5. 

12. 

142:2_4i^  +  15 

132  ELEMENTARY  ALGEBRA 

13..  6x^-x-2.  14.  4y2  +  it3^  +  15. 

15.    6a^-{-Ux-\-6.  16.  20^2  + 13  m -15. 

17.    4a2H-8a  +  3.  18.  6^2^ii^_10. 

19.    1522  +  162  +  4.  20.  8^>2_266-45. 

21.    10^:2+9^+2.  22.  10j92  +  29jo  +  10. 

23.    9  a2  +  18  a  +  8.  24.  28  a2  +  51 «  +  20. 

25.    6^2+112  +  4.  26.  6a;2  +  23a:  +  20. 

27.    21a2+8a-4.  28.  12rr2+ 59a;  + 55. 

29.    16^2  + 2a- 3.  30.  16a:2  +  34a:-15. 

31.    '24:X^+7x-6.  32.  20a^  +  53a:+35. 

33.    22a2  +  27a-9.  34.  6ic2  +  (9  +  2a>  + 3a. 

35.    3rr2  +  (^+6>)^  +  2a.  36.  6^2  +  (2a-9)y-3,a. 

37.    a<?rr2  +  (56?+2a>+2  5.  38.  aba^  +  (a^-lr^^x—€d>. 

39.    aJa^  +  (a^  +  52)a;  +  ^5.  40.  3  <?a;2  +  (6  -  2  <?>  -  4. 

41.  6  aV  —  5  ax —  6. 

42.  6  wwp2  +  (3  7^2  +  2  7i2^jt>  H- mw. 

109.  Sum  and  difference  of  two  cubes.  By  actual  mul- 
tiplication we  have, 

{a  +  h)(a^-ab  +  h^}=a^-^h^ 
and  (a-6)(a2  +  a6  +  ^>2^=a3-63; 

conversely, 

(1)  a8  +  63  =  (a+6)(a2-a&H-62). 

(2)  (^-i^  =  (a-b)(i(^  +  ab-hh^). 

By  use  of  identities  (1)  and  (2),  the  factors  of  any  expres- 
sion which  has  the  form  of  the  sum  or  the  difference  of 
two  cubes  may  be  found.  It  is  obvious  that  one  factor  of 
the  sum  of  the  cubes  of  two  numbers  is  the  sum  of  the  num- 
bers and  that  the  other  factor  is  a  trinomial  which  is  the  sum 
of  the  squares  of  the  two  numbers  minus  their  product ;  also, 


TYPE  PRODUCTS  AND  FACTORS      133 

that  one  factor  of  the  difference  of  the  cubes  of  two  numbers 
is  the  difference  of  the  numbers  and  that  the  other  factor  is 
a  trinomial  which  is  the  sum  of  the  squares  of  the  two  num- 
bers plus  their  product, 

ILLUSTRATIVE   EXAMPLES 

1.  Factor  a^  -}-  8. 

Solution.  a3  -H  8  =  a3  +  (2)8 

=  (a  +  2)(a2-2a  +  4).  [§109,1] 

2.  Factor  a^  -  216  R 
Solution.     o8  _  216  &8  =  a3  _  (6  &)« 

=  (a  -  6  6)(o2  +  6  ab  +  36  fe^).         [§  109,  2] 

3.  Factor  x^  +  y^. 

Solution.  a-«  +  y^  -  {jp-y^r  (y^Y 

=  (:r2  +  2/8)[(x2)2  -  (x2)(|,8)  +  (3^«)2] 

That  is,  a:»  +  y^  =  (x'^  +  y^){x^  -  -^V  +  2/®)- 

4.  Factor  a^  —  y^. 

Solution.  a:«  -  yS  =  (a:«  -  y^){x^  +  /)  [§  106] 

=  (a:  -  .y)(a:2  +  xy  +  2^2)  (^  _,.  y^(^x2_xy  +  y^). 

That  is,  a:®  —  y^  =  (a:  —  y)(x  +  .v)(a:2  _j.  -^y  _|.  y2)(-y2  — a;y+y2y^ 


Factor : 

EXERCISE  53 

1.    m^  +  n^. 

2. 

m^  —  ?l3. 

3. 

63  +  ^. 

4.    2:»-l. 

5. 

r3+l. 

6. 

a3-8. 

7.    1-./3. 

8. 

2/3-0^. 

9. 

2:3+27. 

10.    a3H-216R 

11. 

8  a3  4-  27  63. 

12. 

27a3_l. 

13.     8  53-1-1. 

14. 

^-h 

15. 

27^3  +  64713. 

16.    a^-\-l. 

17. 

a^-1. 

18. 

m^-{-m. 

19,    a^-fa:^. 

20. 

ma^  4-  mab^. 

21. 

16r3«  +  2«*. 

22.    a;6^y6. 

23. 

x^-Sx. 

24. 

a363  _  c^. 

134  ELEMENTARY  ALGEBRA 

25.    rn^  +  S,  26.   w«-8.  27.   a6+27. 

28.   mV-a%^.        29.   r«-27.  30.    2  7w8w  +  128w. 

31.    (a; -2)3  +  1.      32.    l-(^x  +  ^y. 

33.  (a;4-y)3+(a;-y)3. 

34.  Give,  at  sight,  one  factor  of  125  —  (a  -|-  4)^. 

35.  Give,  at  sight,  one  factor  of  (a  +  by  —  (a  —  by. 

36.  Give,  at  sight,  one  factor  of  (2m  —  ny-{-(m  +  2  w)^. 

37.  By  use  of  the  factors  of  a:^  —  y^  [See  Illustrative 
Example  4,  p.  133],  find  four  factors  of  999,999. 

Suggestion.     999,999  =  lO^  -  1. 

Factor  : 

38.    l  +  ^\f.       '  39.  (a-6)  +  Ca3-63). 

40.    a-\-b  +  a^-\-b^.  41.  2(jt?+ ^)+jt>3-f- ^. 

42.    (1+ a;) +3(1 +  2^3).  43.  (^a-\-b-cy-(a-b  +  cy, 

44.    m%*  +  m^n.  45.  1  +  (jt?  +  5'  —  1/. 

46.  m^  —  mn(m  +  w)  +  w^. 

Suggestion,     tw*  —  mn(m  +  w)+  n*  =  (w?*  +  w*)  —  mn(in  -f  n). 

47.  m^  —  2  m^w  +  2  Tww^  —  w,8,  43.    2m3  — 12m2+24w  — 16. 
49.    a8  +  fa25  4.|a62+i63.    50.    a^  ^  a^  Jf.\a-{-^j. 

110.  Special  methods  of  factoring.  There  are  certain 
integral  expressions  whose  factors  may  be  found,  and 
which  are  not  classed  under  any  of  the  preceding  general 
cases  of  factoring. 

ILLUSTRATIVE  EXAMPLES 

1.    Factor  a* +a2  +  l. 

Solution,     a*  +  a2  +  1  =  a*  +  2  a2  +  1  -  a2 

That  is,        a*  +  a*  +  1  =  (a2  +  rt  +  l)(a«  -  a  +  1). 


TYPE  PRODUCTS  AND  FACTORS  135 

Remark.  The  solution  of  example  1  illustrates  the  method  of 
factoring  by  the  aid  of  adding  and  subtracting  the  same  number. 

2.  Factor  a^-{-b^-^  4c^- 4:be-^4:ca-2ab. 

Solution,     a^  +  b^  +  4: c^  -  4:bc  +  4: ca  -  2 ab 

=  a2  +  2a(2c  -  &)  +  (4c2  -  46c  +  f^) 

=  a^  +  2a(2c  -  b)  +  (2c  -  by 

=  (a-h2c-by.  [§105] 

3.  Factor  hc(h  —  <?)  -f  ca(^c  —  a)-\-  ahQa  —  6). 

Solution.  In  order  to  arrange  this  expression  according  to  the 
powers  of  a,  it  is  necessary  to  perform  the  indicated  multiplications  in 
the  last  two  terms.     Then  we  have, 

bc{b  —  c)  -\-  ca(c  -  a)  +  ab(a  —  b)  =  bc(b  —  c)  +  c%  —  ca^  +  a^b  —  ab^ 

=  (P-  c)a2  -  (62  -  c^)a  +  bc(b  -  c) 
=  (6  -  c)  [a2  -(b  +  c)a  +  be] 
=  (b  -  c)(a  -  b)(a  -  c). 

Remark.  Many  expressions  when  arranged  according  to  the 
powers  of  some  letter  are  seen  to  be  factorable.  The  solutions  of 
examples  2  and  3  illustrate  the  method  of  factoring  such  expressions. 

.  EXERCISE   54 

Factor  tlie  following : 

1.  w*+m2+l.  2.    w^  +  mV-hw*. 

3.  l-|-9a^^  +  81a^,  4.    m^  +  m^^  +  n^. 

5.  ;?8+Jt>V  +  ?*-  6.    3^+a^i/^-\-y^, 

7.  a^-\-a^-\-l.  8.    16/  + 4^2^!^ 

9.  x^-lla^t/^-^^^  10.    a4_27a2^  +  5*. 

11.  w*-123m2H- 1.  12.    a:4  +  /_7a;2/. 

13.  a*  -  34  a2  -f- 1.  14.   p^  _  14  jo2^2  ^  ^4. 

15.  2:2  +  4  y2  _j_  ^2  _|_  4  ^2  _  2  2a;  _  4  ^^ 

16.  yz(jf  —  z)-\'Zx(z  —  x)-^xy{x  —  y). 

17.  x^  +  y^+l-\-2y  +  2x-\-2xt/. 

18.  a^(b  -  c)  +  bXc  -  a)  +  (^(a  -  5) . 


136  ELEMENTARY  ALGEBRA 

19.  a\b-o)+b\c-a}+c\a-b). 

Suggestion.  Arrange  according  to  powers  of  «,  remove  a  factor, 
then  arrange  the  remaining  factor  according  to  powers  of  b,  remove 
a  second  factor,  finally  arrange  the  remaining  factor  according  to 
powers  of  c. 

20.  9a*-37a262+4  6*. 

•     21.  4a^-21a^f-^9i/^, 

22.  16a^  +  SQa%^  +  Slb^. 

23.  bc(b^  -  c2)  +  ca((^  -  a2)  ^  ab(a^  -  52). 

24.  a(b^-c^)  +  b((^-a^')  +  c(a^-b^). 

25.  62<?2(62  -  (?2)  ^  cH\c^-  a2)  +  ^2^2(^2  _  ^2)^ 

111.  Summary  of  factoring.  The  identities,  rules,  and 
solutions  of  illustrative  examples  as  given  in  this  chapter 
are  sufficient  to  cover  the  simple  cases  of  factoring  which 
occur  in  elementary  algebra.  The  following  summary  will 
be  of  assistance  in  the  work  of  factoring. 

I.  A%  the  fir  %t  step  in  factoring  an  integral  expression  re- 
move all  numerical  and  monomial  literal  factors. 

II.  Iw factoring  a  binomial  use  one  of  these  identities  : 
1.    a2_j2=^(a_&)(a  +  6). 

3.  a3  +  68  =  (a+6)(a2-a6+62). 

III.  In  factoring  a  trinomial  use  one  of  these  identities: 

1.  ff^  +  2a&-f  &2  =  (a+&)2. 

2.  a2-2a&-|-&2  =  (a-&)2. 

3.  jc2  +  (a4-&)Jc+a&=(x+a)(x+ft). 

4.  ax^-\-hx-\-c  =  ipx  +  q)<irx-¥s). 
6.  (^^(^y^-^l^^ia^+h^y-iaby. 


TYPE  PRODUCTS  AND  FACTORS  137 

IV.    In  factoring  a  polynomial  he  guided  hy  one  or  more 
of  the  following  directions : 

1.  Rearrange  and  group  terms. 

2.  Consider  the  polynomial  the  difference  of  two  squares. 

3.  Arrange  the  polynomial  according  to  the  powers  of 
some  one  letter. 

4.  Consider  the  polynomial  the  square  of  a  polynomial. 

EXERCISE   55  — REVIEW 

Find  the  factors  of: 

1.  2a;+2.  2.  a?-\'bx^. 

3.  ax  4-  ay  —  hx  —  hy.  4.  ah  -f-  he. 

5.  3  2;(a~5)-2^(6-a).  6.  p'^-QA:. 

7.  mn\x  —  y)  +  2  m(x  —  y^n  -\-  m(x  —  y). 

8.  a^^-2x'^-x-2.  9.  a^+^a^-x-^. 
10.  a^-2a;-15.  11.  a^  -  a\ 

12.  64a2  +  144«6  +  81J2.        13.    ^x^-yK 

14.  a2»  -  52.  15.    1  -j-  ac?  -  &(i  -  ahcd. 

16.  (a:2  +  «^2)2  4-(^2  +  2>. 

17.  7i(7i4-l)(w4-2)  +  (7i4-l)(w+ 2)(w  +  3). 

18.  2x^  +  2ax-[-2ac-\-2cx.   19.    2ax-^ay—2hx+Sby 
20.  awi  — 2  5w  +  «n  — 2  5m.      21.    (a  +  l)2— 3(a  +  l). 

22.  a2  -  2(a  -h)-  h\  23.  a^  -  2(a  -h)-  h\ 

24.  a^+aV-  25.  (m  +  w)2— p2, 

26.  49^262^- 64  a2c?2.  27.  49a252^-64a262^. 

28.  25a2+10a+l.  29.  3a2-7a?>4-452 

30.  2^2 -12  a: +  36.  31.  jo22:4+23prc2_,_130. 

32.  a;2+2a;-8.  33.  \2m^ -1  pm^  ^-p^. 

34.  8a;3+27.  35.  4m2w2-(w2  +  w2-;?2)2. 


138  ELEMENTARY  ALGEBRA 

36.    Sa^-h6Sa^-S.  37.    a^^  -  a^ 

38.    a6  H-  62  4.  a  _  1.  39.    ab-b^-a  +  1. 

40.  a^  +  6aP  +  5x+l. 

41.  a2_4^(^^^)  +  3(a;4.t^)2. 

42.  (a2_2a)2  +  2(a2-2a)  +  l. 

43.  a2(^  +  l)2-(a-|-l)2(a  +  2)2. 

44.  (a_6)2_(6_a)(a  +  <?). 

45.  (a-  6)(a  +  <?)-(^-«)(^  +  ^)- 

8  68. 


46. 

7i8  +  W*  +  1. 

47. 

a8_3^25_6a62  +  ^ 

48. 

2^ +68  a; +1092. 

49. 

12a2-a-6. 

50. 

4a^-20a%^-hhK 

51. 

2^  -  19  :c2/  +  9  ^. 

52. 

l-Ux-Bla^. 

53. 

a^«  +  7  a:?^  -  30  ^2. 

54. 

lx^-^lxy  +  12y\ 

55. 

10a2_29a6  +  2162. 

56. 

(^2x-yy-Cx  +  i/y, 

57. 

a*+46^ 

58. 

216-(:c+3)3. 

59. 

a^  +  2ae-b^-2bc. 

60.  c2  4.2a6-a2_52,  61.    a6  4-64c6. 

62.  m^  -  p^m^  -  nhn^  ■\- phi\    63.    14  2^2  +  n^;^  _  15^2. 

64.  x^y^  —  x^—u^y^  +  l.  65.    wi^  +  w^-fl. 

66.  2(a;-3)2+3(a;-3)-2.   67.    a^x- y^- c(y  -  x^. 

68.  (2a-36)2+ll(2a-36)+30. 

69.  a6"  —  <?6»+2  +  c?6«+i. 

70.  aa^  -f-  aa;y  +xz-j-  bxy  +  6^2  -1-^2. 

71.  a^J^-a^c^-b^c^  +  d^. 

72.  w(w -f  Jo)— w(7l+Jt?). 

73.  4a252_(^2  +  52__^>)2. 

74.  a2(a  +  6  +  <?  H-  (^)  +  (6<?c?  +  cda  H-  c?a6  +  a6c). 

75.  a2  +  62+i  +  2a  +  26  +  2a6. 

76.  ^a2  +  i62+^c2-Ja6~^6c  +  ^^a. 


TYPE  PRODUCTS  AND  FACTORS  139 

77.  a%^  +  4  ^>V  +  16  c2a2  -  4  ahH  -  16  h(?a  +  8  ca%, 

78.  xyH  —  y^  +  2 (^  —  2). 

79.  2aa:2_(3^^2>  +  (a  +  2). 

80.  3aa;2+(2a-5>-5(a-l). 

112.  The  remainder  theorem.  When  a  polynomial  in  x 
is  divided  by  a  binomial  ic  —  a,  the  remainder  may  be 
found  by  substituting  a  for  x  in  the  polynomial. 

Thus,  when  ^7?  —  Z  x"^  -\-2x  —  f)\&  divided  by  a;  —  2,  the  remainder 
is  the  value  of  1:x?  —  Zx'^-\-1x  —  h  when  2  is  substituted  for  x ; 
namely, 

2  .  23  -  3  .  22  +  2  .  2  -  5  or  16  -  12  +  4  -  5,  which  is  3. 

Similarly,  when  3  x*  +  2  a:  —  5  is  divided  by  a:  —  1,  the  remainder 
is  3  •  1*  4-  2  •  1  —  5,  which  is  0.  Hence,  3a:*4-2a:— 5  is  exactly 
divisible  by  a;  —  1. 

The  proof  in  the  case  of  the  first  of  the  foregoing  ex- 
amples is  as  follows : 

We  know  that  2a;8-3a;2  +  2a:-5=(a:-2)Q  +  /2,  where  d  and 
R  represent,  respectively,  the  quotient  and  remainder  when  Ix^  -Ztc^ 
+  2  a:  —  5  is  divided  by  (a:  —  2).  Since  one  member  of  this  equation 
is  the  same  polynomial  as  the  other,  the  two  members  are  equal  for 
all  values  of  x.     Substituting  2  for  x,  we  have  2  •  2^  -  3  •  2^  +  2  .  2 

-  5  =  (2  -  2)Q  +  /?  or,  since  (2  -  2)Q  =  0  •  Q  =  0,  2^  -  3  •  2^  +  2  •  2 

—  5  =  i?.     A  precisely  similar  proof  holds  when  any  polynomial-  in  x 
is  divided  by  a:  —  a. 

ILLUSTRATIVE  EXAMPLES 

1.  Find  the  remainder  when  7p'-\-Zx-\-b  is  divided  by 
a:- 3. 

Solution.  i2  =  32  +  3.3  +  5  =  23 

2.  Find  the  remainder  when  ^  -\-^^  —  bx-\-Z\B  divided 
by  a;  +  2. 

Solution.  Since  a:  +  2  =  a;  -(-  2),  we  substitute  -  2  for  a:  in 
the  polynomial ;  hence 

i?  =(_  2)8  +  3(- 2)2  -  5(- 2)+ 3  =  17. 


140  ELEMENTARY  ALGEBRA 

3.  Find  the  remainder  when  2a^— 3a;-h7  is  divided 
by  JT. 

Solution.     Since  the  divisor  may  be  written  a;  —  0,  we  have 
i2  =  20-3.0  +  7  =  7. 

4.  Find  the  remainder  when 

a^-(a-{-2}a^  +  (2a  —  S')x-\-4:a  is  divided  by  x-a. 

Solution.     i2  =  aS  -  (a  +  2)a^  +  (2  a  -  3)a  +  4  a 

=  a8-a8-2a2  +2a2-3a  +  4a  =  a. 

5.  Find  the  remainder  when 

a\h  -  (?)  +  ^K^  -  «)  +  ^(«  -  ^)  is  divided  by  a-b. 
Solution.     The  given  expression  may  be  regarded  as  a  polynomial 
in  a.    Hence,  the  remainder  is  the  value  of  the  polynomial  when  b  is 
substituted  for  a.    Thus, 

R  =  b^(b  -  c)  +  b^c  -b)+  c^(b  -  b) 
=  53  _  52c  4.  52^  _  fts  +  0 
=  0. 
Since  the  remainder  is  zero,  the  polynomial  is  exactly  divisible  by 
a-b. 

113,  The  remainder  theorem  in  factoring.  The  linear 
factors  of  a  polynomial  can  often  be  found  by  an  applica- 
tion of  the  remainder  theorem.  In  order  that  a  poly- 
nomial in  X  should  have  a:  —  a  as  one  of  its  factors,  it  is 
sufficient  that  the  polynomial  should  vanish,  that  is, 
should  become  equal  to  zero,  when  a  is  substituted  for  x. 
This  is  evident  since,  by  the  remainder  theorem,  when 
the  polynomial  is  divided  by  a;  —  a,  the  remainder  is  zero, 
if  the  polynomial  vanishes  when  a  is  substituted  for  x. 

ILLUSTRATIVE  EXAMPLES 

1.    Find  the  factors  oi  a^-\-2x^— 5x  —  6. 

Solution.     If  re  -  a,  where  a  is  an  integer,  is  a  factor  of  ar^  +  2  x- 
-  5  X  -  6,  it  is  necessary  that  the  last  term  -  6  should  be  divisible 


TYPE  PRODUCTS  AND  FACTORS  141 

by  a.     We  therefore  substitute  in  turn  the  different  factors  of  —  6, 
namely,  1,  —  1,  2,  —  2,  3,  —  3,  6,  —  6,  in  the  polynomial. 

When  a;  =  l,  a:8  +  2a:2-5a;-6  becomes  1  +  2-5-6=- 8. 

When  x  =  -l,7?  +  2x^-5x-Q  becomes  -1  +  2  +  5-6  =  0. 

When  x  =  2,3^-\-2x^-5x-Q  becomes  2^  +  2  •  2'^  -  5  •  2  -  6  =  0. 

When  x  =  -2,  3?-\-2x^-bx-Q   becomes  (-  2)3  +  2(-  2)'2 
_5(_2)-6=4. 

When  a:  =  3,  a:3  +  2ar2-5a:-6  becomes  38  +  2  •  3^-5  •  3-6  =  24. 

Whena:  =  -3,  a;«  +  2x2-5a:-6  becomes  (-3)3+2(-3)2-5(-3) 
-6  =  0. 

Hence,  when  3^  +  2  x^  -  bx  —  Q  is  divided  by  a:  —  (-  1),  by  a:  —  2 
and  by  a:  —  (—  3),  the  remainder  is  0,  and  therefore  the  factors  of 
ar3  +  2  a:2  -  5  a;  -  6  are  a:  +  1,  a;  -  2,  and  a:  +  3. 

Question.  Why  is  it  unnecessary  to  substitute  6  and  —  6  for  x 
in  the  given  expression  ? 

2.  Show  that  a;"  —  ^",  where  n  is  a  positive  integer,  is 
exactly  divisible  hj  x  —  y. 

Solution.     When  a:**  —  y«  is  divided  by  a:  —  y,  the  remainder,  JR, 
is  the  value  of  a:"  —  y^  when  y  is  substituted  for  x. 
Hence,  R  =  y^  —  y^  =  0. 

3.  When  is  a:"  +-  ^"  exactly  divisible  by  a;  +  ^? 

Solution.  R  =(—  yY  +  ^,  which  is  0  or  2  y",  according  as  n  is 
an  odd  or  an  even  positive  integer.  Therefore,  when  n  is  an  odd 
positive  integer,  a:**  +  y"  is  exactly  divisible  hjx-\-y  and  when  n  is 
an  even  integer  it  is  not  exactly  divisible  hj  x  -\-  y. 

4.  Show  that  a^ -\- h^  +  (^  —  2>  ahc  is  exactly  divisible  by 
a-{-h-\-  c. 

Solution.  Since  a -^  h  -^  c  =  a -{- h  -  c)  the  remainder,  /?,  of 
the  division  is  the  value  of  a^  +  6^  +  c^  —  3  abc  when  —  6  —  c  is  sub- 
stituted for  a. 

Hence,  R=-(h-\-  c)'  +  6^  +  c^  +  3(^>  +  c)hc 

=  -b^-3I^c-d  bc^  _  c3  +  63  +  c8  +  3  b'^c  +  3  bc^ 
=  0. 

Therefore,  the  division  is  exact. 


142  ELEMENTARY  ALGEBRA 


BXEBOISB  66 

Factor : 

1.    a^-x^-x  +  1. 

2.    2^-Sz'\-2, 

3.     3^-i-X^-X-l, 

4.    a:8_3a;-2. 

5.    a^-6x^-{-llx-6. 

6.   a^  +  5x^-x-5. 

7.    2a^-x^-6x-2. 

Suggestion.     When  two  factors  have  been  found,  the  third  factor 

may  be  obtained  by  dividing  the  given  expression  by  the  product  of 
the  two  known  factors. 

8.    Sa^-2a^-19x-e.  9.    2a^ -\-x^-lSx-\-6. 

10.  a^4-5aj3  +  5a:2_5a._6, 

11.  Show  that  a;2»  —  ^2n^  where  w  is  a  positive  integer,  is 
exactly  divisible  by  a;  +  y. 

Equations  Solved  by  Factoring 

114.  Rational  and  integral  equations  in  one  unknown 
number.  By  transposition,  all  the  terms  of  an  equation 
can  be  brought  to  one  member  of  the  equation.  The  other 
member  then  is  zero.  An  equation  which  contains  one 
and  only  one  unknown  number  is  said  to  be  a  rational  and 
integral  equation  in  one  unknown,  provided  that,  when 
all  the  terms  are  written  in  one  member,  the  polynomial 
which  occurs  in  that  member  is  rational  and  integral  with 
respect  to  the  unknown. 

Thus,  3  a:»  -  I  a;2  +  1 X  -  ^  =  0  is  a  rational  integral  equation. 
Remark.     When  the  word  equation  is  used  in  this  chapter,  it  will 
be  understood  to  refer  to  a  rational  integral  equation  in  one  unknown. 

115.  Degree  of  an  equation.  Any  equation  may  be  put 
in  the  form  ^  =  0,  in  which  A  represents  a  rational  and 
integral  expression  with  respect  to  the  unknown.  Equa- 
tions are  classified  according  to  the  degree  of  the  expres- 


TYPE  PRODUCTS  AND  PACTOPB  143 

sion,  represented  by  A^  with  respect  to  the  unknown 
number.     For  example, 

aa;  +  &  =  0  is  a  linear  equation^  or  an  equation  of  the  first  degree. 

ax^  +  6a:  +  c  =  0  is  a  quadratic  equation^  or  an  equation  of  the 
second  degree. 

ax^  +  6a~^  +  car  +  c?  =  0  is  a  cubic  equation^  or  an  equation  of  the 
third  degree. 

ax^  +  6a;*  +  ca;2  4-  rfa:  4-  e  =  0  is  an  equation  of  the  fourth  degree. 
[§  85,  Remark.] 

116.  Solution  of  an  equation.  An  equation  in  one 
unknown  is  said  to  be  solved  when  all  of  its  roots  have 
been  found. 

117.  Roots  of  an  equation  found  by  factoring.     If  one 

member  of  an  equation  is  zero,  the  roots  of  the  equation 
may  be  found  easily,  provided  that  the  polynomial  in  the 
other  member  can  be  expressed  as  a  product  of  factors, 
each  one  of  which  is  of  the  first  degree  in  the  unknown 
number.  This  important  method  of  solving  an  equation 
is  applied  and  explained  in  the  illustrative  examples  which 
follow. 

ILLUSTRATIVE  EXAMPLES 

1.    Solve  the  equation  ^x^-\- ^x=2x^ -  Zx'\-^^. 

Solution.  3  a;2  +  5  a:  =  2  a;2  -  3  a;  +  33.  (1) 

Transposing,  3  a;2  +  5  a;  -  2  x2  +  3  a:  -  33  =  0.  (2) 

Combining,  a;2  -|-  8  a;  -  33  =  0.  (3) 

Factoring,  {x  -  3)(a:  +  11)  =  0.  (4) 

Notice  here,  that  the  first  member  of  equation  (4)  is  a  product  of 
factors  and  that  a  product  cannot  be  equal  to  zero  unless  at  least  one 
of  its  factors  is  zero.  Hence,  any  value  of  x  which  satisfies  equation 
(4)  must  cause  at  least  one  of  the  factors  of  the  first  member  to  vanish. 
Moreover,  any  value  of  x  which  causes  either  of  the  factors  to  vanish 
will  satisfy  the  equation.  Therefore,  the  required  roots  are  found  by 
equating  each  of  the  factors  to  zero. 


144  ELEMENTARY  ALGEBRA 

Therefore,  a;  -  3  =  0.                                            (5) 

Whence,  x  =  d.                                          (6) 

Also,  a;  +  11  =  0.                                            (7) 

Whence,  x  =  -  11.                                      (8) 

That  is,  the  roots  of  3  a;^  +  5  a;  =  2  a;^  -  3  a;  +  33  are  3  and  -11 

2.  Solve  the  equation  ^a^ -2a^- ^  x-\- 2  =  0. 

Solution.                                    3  x8  -  2  a:2  -  3  a:  +  2  =  0.  (1) 

Grouping  terms,                    (3  a:8  -  2  a:2)  -  (3  a;  -  2)  =  0.  (2) 

Factoring  first  term  of  (2),    a;2(3  a;  -  2)  -  (3  a;  -  2)  =  0.  (3) 

Factoring,                                        (3  a;  -  2)  (a?^  - 1)  =  0.  (4) 

Factoring  completely,          (3  a;  -  2)  (a;  -  l)(a:  +  1)  =  0.  (5) 

Equating  (3  x  -  2)  to  0,                                  3  a;  -  2  =  0.  (6) 

Equating  (a:  -  1)  to  0,                                        a;  - 1  =  0.  (7) 

Equating  (a;  +  1)  to  0,                                        a;  + 1  =  0.  (8) 

Solving  (6),                                                                  a;  =  |.  (9) 

Solving  (7),                                                                a;  =  L  (10) 

Solving  (8),                                                                   a:  =  -  1.  (11) 
That  is,  the  roots  of  3  x^  -  2  a;^  -  3  a:  +  2  are  1,  —  1,  and  ^. 

3.  Solve  the  equation  a^  —  Sx^-{-16  =  0. 

Solution.  a;4  -  8  a:2  +  16  =  0.  (1) 

That  is,  (a:2  -  4)2  =  0.  (2) 

Factoring,  [(a:  -  2)(a:  +  2)]2  =  0.  (3) 

That  is,     (x  -  2)(x  -  2)(x  +  2)(ar  +  2)  =  0.  (4) 

Equating  each  factor  to  0,  a:  — 2  =  0,  a:  — 2  =  0,  a;  +  2  =  0,  x  +  2  =  0. 
Solving  simple  equations,         a:=  2,       a:  =  2,        a:  =  —  2,    a:  =  —  2. 
That  is,  the  roots  of  a;^  -  8  a:2  +  16  =  0  are  2,  2,  -  2,  and  -  2. 
Note.     The  equation  has  four  roots,  two  pairs  of  equal  roots ;  that 

is,  as  many  roots  as  there  are  linear  factors. 

Remark.     The  student  should  carefully  check  all  roots  obtained 

from  the  solutions  of  illustrative  examples  1,  2,  and  3. 

4.  Solve  Qa^-llx^-S5x=0. 

Solution.  6  a;8  -  11  a;2  -  35  a;  =  0. 

Factoring,  a:(3  a:  +  5)  (2  a:  -  7)  =  0. 

Equating  each  factor  to  0,        a:  =  0,   3  a:  +  5  =  0,   2  a:  —  7  =  0. 

Solving  simple  equations,  a:  =  0,   a:  =  —  J,   a:  =  |. 

That  is,  the  roots  of  6  a:8  -  11  a:2  -  35  x  =  0  are  0,  -  J,  and  J. 


TYPE  PRODUCTS  AND  FACTORS  145 

From  the  solutions  of  the  foregoing  illustrative  ex- 
amples, the  following  rule  for  solving  an  equation  by 
factoring  may  be  inferred : 

Rule.  Transpose  all  the  terms  to  one  member  of  the  equa- 
tion^ factor  the  resulting  expression  into  its  linear  factors^ 
equate  each  factor  to  zero^  and  solve  the  resulting  simple 
equations. 

Note.  In  solving  equations  by  factoring,  care  should  be  exer- 
cised to  bring  all  terms  to  one  member  of  the  equation.  The  following 
is  an  example  of  an  error  which  is  the  direct  result  of  disregarding 
this  practice. 

Solve  the  equation  (2  a;  +  3)  (a:  -  1)  =  (a:  +  2)(a:  -  1). 

Incorrect  Solution.  (2  a:  +  3)(x  -  1)  =  (ar  +  2) (a:  -  1).  (1) 

Dividing  both  members  of  (1)  by  (x  —  1), 

2  a:  +  3  =  X  +  2.  (2) 

Transposing  and  combining,  a:  +  1  =  0.  (3) 

Solving,  X  =  —  1.  (4) 

Correct  Solution.  (2  a:  +  3)(a;  -  1)  =  (a:  +  2) (a:  -  1). 

Transposing, 

(2a;  +  3)(a:  -  l)-(x  +  2)(a:  -  1)=  0.  (1) 

Factoring,  (x  -  1)[(2  a:  +  3)  -  (a:  +  2)]  =  0.  (2) 

SimpUfying,  (x  -  l)(a:  +  1)  =  0.  (3) 

Equating  each  factor  to  0,  a:  —  1  =  0,   a:  +  1  =  0. 

Solving  simple  equations,  a;  =  1,   a:  =  —  1. 

That  is,  the  roots  of  (2  a:  +  3)  (a:  -  1)  =  (a:  +  2)  (a;  -  1)  are  +  1  and 
-1. 

The  error  in  the  incorrect  solution  arises  from  dividing  both  mem- 
bers of  equation  (1)  by  a  factor  which  contains  the  unknown  num- 
ber, and  which  vanishes  when  x  has  the  value  1. 

The  equations  (2  a:  -f-  3) (a:  -  1)  =  (x  +  2) (a: - 1)  and  2x  +3  =  x-\-2 
are  not  equivalent,  the  first  equation  having  a  root  which  is  not  a 
root  of  the  second.  In  solving  equations,  every  transformed  equation 
or  set  of  equations  wMch  occurs  in  the  solution  must  be  equivalent  to 
the  original  equation  [§74]. 


146  ELEMENTARY  ALGEBRA 

EXERCISE  57 

Solve  the  following  equations  and  check  the  roots : 

1.  a?(a:-2)=0.  2.  r» (3 a; -h  2)  =  0. 

3.  (2a:+l)(3a;-l)  =  0.  4.  a^-2x-^l  =  0, 

5.  ar2-3a;H-2=0.  6.  a^-9x  +  20  =  0, 

7.  2^2  + 2a; -3  =  0.  8.  rrZ  +  3^;  _  IQ  =  0. 

9.  a;2-13a;  +  42  =  0.  10.  a^*- 6a:  -  55  =  0. 

11.  a^-5a;+6  =  0.  12.  a^^- 4a:- 21  =  0. 

13.  a^-\-x-SO  =  0,  14.  a:2_7^^io^O. 

15.  a^-Ux-15=:0.  16.  2a:2-3a:-2  =  0. 

17.  3a;2-h2a:-8  =  0.  18.  4a:2  -  3  a:  -  85  =  0. 

19.  3a^2-5a:-12  =  0.  20.  4a:2  _  3^  _  45  ^  0. 

21.  5a^  +  a:-6=0.  22.  4x^^lx-Ul  =  0. 

23.  7a:2_5a:-78  =  0.  24.  11  a^^- 13a;  4- 2  =  0. 

25.    15a^i  +  2a;-56  =  0.  26.    13ar2- 9a;  -  414  =  0. 

27.  3a;2^2a:  +  5  =  5a;2_3^_2. 

28.  a;(a;+l)  =  (2a;  +  l)(a;  +  l). 

29.  a;(a;H-l)(a;  +  2)=a;(2a;  +  3)(a;-t-l). 

30.  (2a;2_3^_,.i)2_(^_  1)2  =  0. 

Highest  Common  Factor— Lowest  Common  Multiple 

118.  Highest  common  factor.  The  highest  common 
factor  (H.  C.  F.)  of  two  or  more  integral  expressions  is  the 
integral  expression  of  the  highest  degree,  with  greatest 
numerical  coefficient,  which  exactly  divides  each  of  them. 

Thus,  the  H.  C.  F.  of  4  a%»  and  6  a%^  is  evidently  2  a%^. 

119.  Greatest  common  diyisor  (G.  CD)  in  arithmetic. 
In  arithmetic,  the  greatest  common  ^ivisor  of  two  or 
more  numbers  may  be  found  by  expressing  each  of  them 


TYPE  PRODUCTS  AND  FACTORS  147 

as  the  product  of  powers  of  its  different  prime  factors, 
and  then  taking  the  product  of  the  common  prime  factors 
of  the  numbers,  giving  to  each  common  prime  factor  the 
least  exponent  which  it  has  in  any  of  the  numbers. 

ILLUSTRATIVE  EXAMPLE 

Find  the  greatest  common  divisor  of  180,  252,  and  270. 
Solution.  180  =  22  x  3^  x  5 

252  =  32  X  7  X  22 

270  =  2  X  38  X  5 
G.  C.  D.  of  180,  252,  and  270  =  2    x  32,  or  18. 

120.  Highest  common  factor  of  monomials.  The  high- 
est common  factor  of  two  or  more  literal  monomials  can, 
obviously,  be  found  by  inspection. 

ILLUSTRATIVE  EXAMPLE 

Find  the  highest  common  factor  of  12  a^^c^,  18  cfil^(^^ 
and  24  a%'^c. 

The  greatest  number  which  will  exactly  divide  12, 18,  and  24  is  6. 
The  highest  power  of  a  which  will  exactly  divide  a,  a^,  and  a*  is  a. 
The  highest  power  of  h  which  will  exactly  divide  62,  &»,  and  6*  is  h^. 
The  highest  power  of  c  which  will  exactly  divide  c*  and  c  is  c. 
Evidently,  the  required  H.  C.  F.  is  6  ab^c. 

From  the  above  illustration,  we  have  the  following: 

Rule.  To  find  the  highest  common  factor  of  two  or  more 
monomials,  multiply  the  product  of  the  lowest  powers  of 
their  common  literal  factors  hy  the  greatest  common  divisor 
of  their  numerical  coefficients. 

Note.  The  numerical  coefficient  in  the  highest  common  factor 
is  taken  as  positive. 

Thus,  the  H.  C.  F.  of  -  4  a^h  and  6  a/>2  jg  regarded  as  2  ah  and 
not  —  2  ah. 

Remark.  When  the  greatest  common  divisor  of  the  numerical 
coefficients  of  two  or  more  monomials  cannot  be  readily  seen,  it  may 
be  found  as  in  section  119. 


148  ELEMENTARY  ALGEBRA 

EXERCISE  58 

Find  by  inspection  the  highest  common  factor  of : 

1.  a%^  and  a¥.  2.    3  x^yz  and  12  s^y. 

3.  6  a'^h^c  and  4  a^h\  4.    a!^c,  a^c^,  and  a%<?. 

5.  2  a^b'^c^  4  a^ftc^  and  6  aJ^.      6.    (a:  +  y)z  and  (a;  —  y)2. 

7.  —  2(a  +  6)a:^  and  —  2(a  +  6):r2. 

8.  (a  +  ^) V  and  (a  +  ft)  V. 

9.  2a25(c  +  e^)2,  4a52(c  +  (^),  and  10ah{c^-dy, 

10.    a;(a;-l)(a;-2),  a;(2:4-l)(a:-l),  and  3a:(a;-l)(a;-2). 

121.  Highest  common  factor  of  polynomials  by  factoring. 

Expressions  which  are  completely  factored;  i.e.  each  of 
which  is  expressed  as  a  product  of  powers  of  its  prime  fac- 
tors, are  in  the  form  of  monomials,  and  their  highest  com- 
mon factor  may  be  found  by  inspection,  as  in  exercise  58. 

ILLUSTRATIVE  EXAMPLE 

Find  the  highest  common  factor  of  18  a;^  -|-  15  a;  —  18 
and  36  a:2  +  78  a:  +  36. 

Solution.     18  a;2  +  15  a:  -  18  =  3(2  a;  +  3)(3  a:  -  2) 
36  x2  +  78  X  +  36  =  6(2  x  +  3)(3  x  +  2). 

Therefore,  the  required  highest  common  factor  is  the  H.  C.  F.  of 
3(2  a:+3)(3  x-  2)  and  6(2  a:  +  3)(3  a;  +  2),  which  is  evidently  3(2  a:+3). 

Therefore,  to  find  the  highest  common  factor  of  two  or 
more  polynomials  which  can  he  readily  factored.,  the  method 
of  procedure  is  to  factor  each  polynomial  completely.,  thus 
changing  each  into  the  form  of  a  monomial,  and  then  to  find 
the  highest  common  factor  of  the  monomials  by  inspection. 

EXERCISE  59 

Find  the  highest  common  factor  of  : 

1.  (2a:4-4)(a;+2)  and  (3a:  +  6)(a;  +  2). 

2.  x^-^'^x  and  2 a; +  6, 


TYPE  PRODUCTS  AND  FACTORS  149 

3.  x^+2x-{-landa^-Sx-4:. 

4.  2^2  +  3  2;  _  10  and  a^  -  5  2;  +  6. 

5.  a%  +  ab^2inda^+a%. 

6.  3^  —  x^  and  2^  -\-a^. 

7.  aHSft^and  2a^-\-^ab. 

8.  a^  —  y^  amd  x!^ -{- 3^ -\- 1. 

9.  a3  ^  53  and  (2  a  +  3  5)2  -  (a  +  2  5)2. 

10.  3  (a  +  5),  6  a2  +  6  a5,  and  2  a^  +  2  a^. 

11.  ax  —  ay  -\-hx  —  by  and  aa:  +  5a;  +  5y  +  ay. 

12.  a:2_9^  ^_6a;_^9^  and  2a^2_5a,_3^ 

13.  1155  a%,  910  a52,  and  595  ah. 

14.  a2-52_c2+25cand  a2_52_^^^2ac. 

15.  4  a;2  +  12  a;  +  9,  4  a:2  -  9,  and  6  a:3  _^  13  2,2  ^  5  ^, 

16.  a2+52  +  c2_25c-2ca  +  2a5and  a^-\-b'^-(^  +  2ab. 

17.  iK3-2a^  +  a;-2auda:3_  3a:2  +  ^_3. 

18.  a;2-3a;H-2  anda;3_^5^2_3a._3^ 

Suggestion.  The  first  expression  is  the  product  of  two  factors. 
Find  whether  the  second  expression  is  exactly  divisible  by  one  or  both 
of  these  factors. 

19.  x^  +  ix-5  and  a;^  +  3  a;2  -  9  a;  +  5. 

20.  x^  —  X—  6  and  a:^  +  a;2  —  9  rr  —  9. 

122.  Lowest  common  multiple.  The  lowest  common 
multiple  (L.  C.  M.)  of  two  or  more  integral  expressions, 
the  numerical  coefficients  of  which  are  integers,  is  the  in- 
tegral expression  of  lowest  degree  with  least  numerical 
coefficient  which  is  exactly  divisible  by  each  of  them. 

Thus,  the  L.  C.  M.  of  2  a^b*  and  4  a^b^  is  evidently  4  aSJ*. 

123.  Least  common  multiple  (L.  C  M.)  in  arithmetic. 

In  arithmetic  the  least  common  multiple  of  two  or  more 
numbers  may  be  found  by  expressing  each  of  them  as  the 


150  ELEMENTARY  ALGEBRA 

product  of  powers  of  its  different  prime  factors  and  taking 
the  product  of  all  the  different  prime  factors  of  the  num- 
bers, giving  to  each  different  prime  factor  the  greatest 
exponent  which  it  has  in  any  of  the  numbers. 

ILLUSTRATIVE  EXAMPLE 

Find  the  least  common  multiple  of  90,  189,  and  300. 

Solution.  90  =  2  X  32  X  5 

189  =  38  X  7 
300  =  22  X  3  X  52 
L.  C.  M.  of  90, 189,  and  300  =  22  x  38  x  5^  x  7,  or  18,900. 

124.  Lowest  common  multiple  of  monomials.  The  low- 
est common  multiple  of  two  or  more  literal  monomials 
can,  obviously,  be  found  by  inspection. 

ILLUSTRATIVE  EXAMPLE 

Find  the  lowest  common  multiple  of  6  a^h(^^  8  ahc^^  and 
12  aW(^, 

Solution. 

By  inspection  it  is  readily  seen  that : 
The  least  number  which  will  contain  6,  8,  and  12  is  24. 
The  lowest  power  of  a  which  will  contain  a^,  a,  and  a'  is  a*. 
The  lowest  power  of  b  which  will  contain  h  and  b^  is  ft*. 
The  lowest  power  of  c  which  will  contain  c^  and  c^  is  c*. 
Evidently,  the  required  L.  C.  M.  is  24  amc\ 

From  the  above  illustration,  we  have  the  following: 

Rule.  To  find  the  lowest  common  multiple  of  two  or  more 
monomiaU^  multiply  the  product  of  the  highest  powers  of  their 
different  literal  factors  hy  the  least  common  multiple  of  their 
numerical  coefficients. 

Remark.  When  the  lowest  common  multiple  of  the  numerical 
coefficients  of  two  or  more  monomials  cannot  be  readily  seen,  it  may 
be  found  as  in  section  123. 


TYPE  PRODUCTS  AND  FACTORS.  151 

EXERCISE  60 

Find,  at  sight,  the  lowest  common  multiple  of: 
1.    2  a^h  and  3  ac^,  2.    ahc^  3  a^c^  and  5  aft^. 

3.    5  a;,  3  y,  and  2z,  4.    21  :^y,  %\yh,  and  22%. 

5.  Qx^yz^  ISa^z,  and  \%xyz^, 

6.  2  mVp\  3  a^TTjp^  and  6  a6<?. 

7.  a;(a;  —  1)  and  y(x  —  1). 

8.  a;2(a;  —  1)  and  xy(x  —  l)^. 

9.  x^(x  -  iy(x  +  3)  and  y\y  -  r)\x  +  3). 
10.    17  x^y\x  +  yy  and  10  a:8«^(2J  +  yy. 

125.  Lowest  common  multiple  of  polynomials  by  factor- 
ing. Expressions  which  are  completely  factored,  that  is, 
each  of  which  is  expressed  as  a  product  of  powers  of  its 
prime  factors,  are  in  the  form  of  monomials,  and  their 
lowest  common  multiple  may  be  found  by  the  rule  of  sec- 
tion 124. 

ILLUSTRATIVE   EXAMPLES 

1.  Find  the  lowest  common  multiple  of  x^'\'4iX  and 
3a;+12. 

Solution.  a^*  H-  4  X  =  x{x  +  4). 

3a:  +  12  =  3(a;  +  4). 
Therefore,  the  required  L.  C.  M.  =  the  L.  C.  M.  of  x(x  +  4)  and 
3(a;  +  4),  or  3  x{x  +  4). 

2.  Find  the  lowest  common  multiple  of  3  a:^  __  27  a;  +  60, 
a;3— 5a^4-a:  —  5,  and  a^ —  4:3^  + x  —  i. 

Solution.     S x^  -  27 X  +Q0  =  S(x  -  ^)(x  -  5). 

x^-5xi-\-x-6=(x-  5)(x^  +  1). 

x»-ix^  +  x-^  =  (x-  4)(a^  +  1). 
Therefore,  the  required  L.  C.  M.  =  the  L.  C.  M.  of  3(x  -  4)(x  -  5), 
(x  -  5)(a:2  +  1),  and  (x  -  4)(a^^  +  1),  or  3(a;  -  i)(x  -  o)(x^  +  1). 


152  ELEMENTARY  ALGEBRA 

3.  Find  the  lowest  common  multiple  of  a:^  -{-x^  —  6x 
and  a^  —  6  x^  -^  11 X  —  Q, 

Solution.     x»  -\-  x^  -  Q  X  =  x(x  +  S)(x  -  2). 

It  is  now  necessary  to  find  whether  or  not  a:*  —  6  x^  -f  11  a;  —  6  is 
exactly  divisible  by  any  of  the  factors  oi  t^  -\-  x^  —  Q  x.  By  actual 
division  a:^  —  6  x^  +  11  a;  —  6  is  found  to  have  a:  —  2  as  a  factor ;  hence, 
x9-Qx^  +  nx-Q  =  (x-l)(x-2)(x-  3). 

Therefore,  the  required  L.  C.  M.  =  the  L.  C.  M.  of  x(x  +  3)(a;  -  2) 
and  (x  -  l)(x  -  2)(x  -  3),  or  a;(a;  -  l)(a:  -  2)(a:  -  3)(a;  +  3). 

EXERCISE  61 

Find  by  factoring  the  lowest  common  multiple  of : 

1.  a%  +  ab^aLnda^-a%. 

2.  a*  —  a^  and  a^  —  a. 

3.  a^+6a;and4a;4-24. 

4.  (2a;  +  4)(a;4-2)  and  (3rr  +  6)(a;  +  2). 

5.  a2  _  52  and  a3  _  53. 

6.  x^  —  1  and  s^—1. 

7.  a3  4-  h^  and  a*  +  a%^  +  6*. 

8.  a2-62and  (a-i)(a2  +  62). 

9.  x^  +  5x-68Lnda^-dx  +  2. 

10.  x^-\-  bx  —  14  and  x^  —  5x-{-6. 

11.  x^-f,x^-}-xy-2fsindx^-\-Sxt/-\-2f. 

12.  122:2-18a:y +  453/2  and  18a:2-33a;y-30^2. 

13.  3^—2x^-\-xsinda^  +  x^—x  —  l. 

14.  a^  —  ^y,  y^  —  xy,  and  a^—Zx^y-\-xy^  +  y^. 

15.  (x  +  y)2  -  xy,  7^  -  y^,  and  a:^  4.  a^?^  +  xy^. 

16.  (a  +  6)2  -  c2,  (a+c)2- J2,  and  (6  +  0^  -  «^- 


CHAPTER   V 
FRACTIONS 

126.  Definition.  A  fraction  in  algebra  is  the  quotient 
of  two  numbers  or  expressions.     The  expression  -  means 

0 

a-*-6  (§  6,  Remark).  A  fraction,  however,  is  usually  re- 
garded ^s  an  indicated  division.  The  dividend,  a,  is  called 
the  numerator,  and  the  divisor,  b,  is  called  the  denominator. 
The  numerator  and  denominator  are  called  the  terms  of 

the  fraction.     ^  ^^  ^^^^  ^  ^^^^  6  or  a  divided  by  h. 

0 

127.  Laws  governing  algebraic  fractions.  Algebraic 
fractions  are  subject  to  the  same  laws  as  arithmetical  frac- 
tions. This  is,  in  part,  a  direct  consequence  of  the  fact 
that  both  kinds  of  fractions  remain  unchanged  in  value 
when  both  numerator  and  denominator  are  multiplied  or 
divided  by  the  same  number  (excepting  zero).  For  arith- 
metical fractions  this  important  principle  is  established  in 
arithmetic.     It  may  be  proved  in  algebra  as  follows : 

Let  7  denote  any  fraction,  and  m  the  number  by  which  its  terms 

0 

are  to  be  multipHed.  Representing  the  quotient  of  a  divided  by  h  by 
q  we  have, 

Since  the  dividend  is  equal  to  the  product  of  the  divisor  and  the 
quotient,  a  =  bq.  (2) 

163 


154  ELEMENTARY  ALGEBRA 

Multiplying  both  members  of  identity  (2)  by  m, 

am  =  bmq.  (3) 

Dividing  both  members  of  identity  (3)  by  bm, 

"£=*•  (*) 

From  identities  (1)  and  (4),  we  have, 

b     bm 

Again,  ^  may  be  obtained  from  —  by  dividing  both  terms  of  — 
0  bm  bm 

by  m,  and  from  identity  (5)  the  value  of  the  fraction  remains  un- 
changed ;  that  is,  we  may  write 

^  =  ?.  (6) 

bm     b 

From  identities  (5)  and  (6)  we  have  the  following 
principle : 

Multiplying  or  dividing  both  terms  of  a  fraction  by  the 
same  number  (zero  excepted)  does  not  change  its  value. 

Note.  The  denominator  of  a  fraction  cannot  be  zero.  The  expres- 
sion -  has  no  meaning,  and,  therefore,  does  not  represent  a 'number. 

In  other  words,  division  by  zero  is  excluded.  Care  should  be  exer- 
cised in  assigning  numerical  values  to  letters  to  see  that  the  values 
assigned  do  not  cause  the  denominator  of  a  fraction  to  vanish. 

128.  Signs  affecting  a  fraction.     The  sign  of  a  fraction 

is  the  plus  or  minus  sign  before  the  fraction. 

—  2 

Thus,  in  the  expression  -i ,  the  sign  which  stands  first  is  the 

+  3 
sign  of  the  fraction. 

There  are,  therefore,  three  signs  which  affect  a  fraction  ; 
namely,  the  sign  of  the  fraction,  the  sign  of  the  numerator, 
and  the  sign  of  the  denominator.  The  signs  of  the  nu- 
merator and  denominator  combine  according  to  the  rule 
for  signs  in  division. 


FRACTIONS  155 

Thus, 

±|  =  +  |.  (1)  ^  =  +  |.  (2) 

-8=-i-        («)  fl=-i-        (*> 

In  algebraic  symbols, 

±^  =  -l-  (3)     •  =^  =  -r  (*) 

—   00  -\-   0  0 

By  comparing  equation  (1)  with  equation  (2)  and  equa- 
tion (3)  with  equation  (4),  it  is  evident  that 

I.  The  signs  of  both  terms  of  a  fraction  may  he  changed 
mthout  altering  the  value  of  the  fraction. 

Thus,  ±^  =  :::^and  ^=^. 

+  b       -b  -b       +  b 

II.  Any  two  of  the  three  signs  affecting  a  fraction  may  be 
changed  without  altering  the  value  of  the  fraction. 

The  foregoing  statement  is  evident  from  (I)  and  from 
the  following : 

Changing  the  sign  of  the  fraction  and  the  sigti  of  the 
numerator  in  (1),  we  have 

Changing  the  sign  of  the  fraction  and  the  sign  of  the 
denominator  in  (1),  we  have 


^51=+'  + 


!)=+!•  c^) 


Comparing  identities  (1),  (2),  (3),  we  have 

—  ^^—r  =  +  ^^  =  -\-  ^— ,  since  each  is  equal  to  -|-  ^ 
4-0+6—6  6 


156  ELEMENTARY  ALGEBRA 

From  I  and  II  it  is  evident  that 

in.  A  fraction  may  he  written  in  at  least  four  ways  with- 
out changing  its  value. 

Thus,  +±i^  =  +  Zl^  =  _±^  =  _ZL^. 

-fft  -h  -  b  +6 

In  like  manner, 

I     ^  ~  -'   =  \     -  ^  +  y    -     -  ^  +  .y  ^        X-  y 

X  —  y  —  z  —  X  +  y  +  z  x  —  y  —  z  —  x  -\-  y  +  z 

Remark.  When  no  sign  is  written  before  a  fraction,  +  is  under- 
stood. 

Thus,  -  means  +-• 
b  b 

129.  Change  of  signs  of  factors  in  the  terms  of  a  frac- 
tion. To  change  the  sign  of  one  factor  of  an  expression 
is  equivalent  to  multiplying  that  expression  by  —  1. 
Therefore,  when  either  or  both  terms  of  a  fraction  are  ex- 
pressed as  a  product  of  factors,  the  signs  of  an  even  num- 
ber of  these  factors  may  be  changed  without  altering  the 
value  of  the  fraction ;  but  if  the  signs  of  an  odd  number 
of  them  are  changed  the  sign  of  the  fraction  must  be 
changed  in  order  that  its  value  may  not  be  changed. 

Thus,  (g  -  b)(c  -  d)  ^  (b  -  aXd  -  c)  ™,     ^-. 

(x  -  y)iz  -w)      {x  -  y){z  -  w) 

=  -(«  -h){c-d)  [Why?] 

{X  -  y)(ia  -z)  I       y  ^ 

ILLUSTRATIVE  EXAMPLE 

Without  altering  the  value  of  the  fraction  — ^~^,  ex- 
press it  in  a  form  in  which  each  of  the  three  signs  affecting 
it  is  plus. 

Solution.  Changing  the  sign  of  the  fraction  and  the  sign  of  the 
denominator,  we  have 

y  -  X  _y -  X 
-2a        2a 


FRACTIONS  157 


EXEBCISE   62 

Tell,  at  sight,  which  of  the  statements  in  examples  1-15 
are  true. 

1.    +£  =  +^^. 

y       -y 

3.     +E  =  -I1^. 

y       -y 

5.  --^  =  -z:£. 
-y        y 

'  X  X 


2. 

y       -y 

4. 

X         —X 

y     y 

6. 

X             —  X 

-y    -y 

8. 

X                —  X 

9. 


10. 


11. 


12. 


13. 


X-y  y-x                               x-y      y-x 

X     _  —  a; 

x-y  x-y 

a  a 


(a  -  6)(e  -  df)      (h  -  a){d  -  <?) 

a —  a 

(a  _  h)(c  -d)~  (h-a)(c-d) 
a  a 


(a  _  hXe  -d)      (5  -  a)(c  -  d) 
a-\-b  ^a  —  h 
c—  d      d  —  c 


a-\-  b  _  —  a  —  h  a-\-h  __  _  —  a  —  h 

c  —  d        d  —  c  .                    c  —  d            c  —  d 

Without  altering  the  value  of  the  fractions  in  examples 

16-30,  express  each  in  a  form  in  which  the  three  signs 
affecting  it  are  plus. 

16.    =^.  17.    '^^^. 

—  0  —  c 

18. .  •     19. ^. 

y  —a 


158  ELEMENTARY  ALGEBRA 

a(h  —  g) 


90 

—  a  —  h-\-  c 

ah 

22. 

h-\-c 

c  —  a 

24 

^(a^h)(x^y) 

x-\-  a 

91*. 

~  a 

y(a-5) 

23. 

y-x 

25. 

a-\-h 
-c-d 

on 

« 

5(c  —  a)  (c  —  a)6 

28. 


29.    _JL!lA_.  30.     --(^-^). 

—  (m  —  w)  —  (r  —  «) 


Lowest  Terms 

130.  Numbers  or  algebraic  expressions  prime  to  each 
other.  Two  numbers  in  arithmetic  or  two  algebraic  ex- 
pressions are  said  to  be  prime  to  each  other  when  their 
only  common  factor  is  1. 

131.  Reduction  of  fractions  to  lowest  terms.  A  fraction 
is  said  to  be  in  lowest  terms  when  its  numerator  and  de- 
nominator are  prime  to  each  other. 

It  was  shown  in  section  127  that  both  terms  of  a 
fraction  may  be  divided  by  the  same  number  without 
changing  the  value  of  the  fraction.  Hence,  we  have  the 
following: 

Rule.  To  reduce  a  fraction  to  lowest  terms,  cancel  all 
factors  common  to  the  numerator  and  denominator ;  that  is, 
divide  both  terms  of  the  fraction  by  their  highest  common 
factor. 


FRACTIONS  169 


ILLUSTRATIVE  EXAMPLES 

1.    Reduce  ^^    „,  ^ ,  to  lowest  terms. 

28  a%M 

Solution.      To  reduce   the  given   fraction  to   lowest  terms,  we 
divide  its  numerator  and  its  denominator  by  their  H.  C.  F.,  which  is 

4  ahcH. 

36  ah'^chJ^  _  36  ah'^cM'^  -^  4  ahc^d  ^  9  hcP 
*'*  28  a^bcH  ~  28  a%cH  -f-  4  ahcH  ~  7  a^^ ' 


2.    Reduce  -r -—  to  lowest  terms. 

a^4-3a:-  18 

Solution.  ^^-9        ^(x-3)(.  +  3)^(^j^, 

a:2+3jr-18      (x-3)(x  +  6)       (j:  +  6) 

Note.  In  practice  it  is  customary  to  separate  the  numerator  and 
the  denominator  into  their  prime  factors  and  cancel  the  factors 
common  to  both. 

^,  3  m^  -3m         _  a>#»{77r=n)  (m  +  1)  ^ m  +  1 

^^'  6  7«5  +  6  m*  _  12  m8  ~  jS.**a(rrr^ri)  (m  +  2)  ~  2  y«2(m  +  2) ' 
2^2 

Remark.  Care  should  be  exercised  not  to  cancel  a  common  term  of 
the  numerator  and  denominator  of  a  fraction  when  they  are  poly- 
nomials. 

EXERCISE  63 

(Solve  as  many  as  possible  at  sight.) 
Reduce  the  following  fractions  to  lowest  terms  : 

^.  2.    — .  3.    ?^. 

xz  4:  be  xyt 

^      -7^  .       12a8 


1. 


a" 


c^  7^  -16  a^ 


^^      - 12  mhiY  9p^qr  UmV 


160  ELEMENTARY  ALGEBRA 


13.  ^m.^ 


2irRH  UttB^ 

xi/{^  +  zy  (a:  H-  z) '  x"" 

17.    —  •  18.     — ^—  • 

2  ^n—lym+2  jf^+lym—l 

19.     ^ •  20.     i^^ 

^,        -35a2«-352p  ^2_52 

21.     •  22.     • 


23.     xr_J — £_.  24. 

25.     ^- 4 26.       ^        ^ 


(a;-l)(a^+l)  46  +  4a 

2^     a2  +  2«a;  +  2^^  28.      ^'-y' 
a^  -]-a^x      '  iy  —  ^Y 

29.     •  30.    ^- ^  • 

53  _  ^8  (f—p^ 

31.        ^'  +  ^     .  32.  ^'"^^ 


2a5  +  26a:  a:2_8^^16 

a^-2xy^-y^  a^  +  h^ 

^^     ax-^bx  +  ay-^  by  ^^  xy 

'    ax-^-ay  —  bx—by  ^y  +  xy'^ 

37.     <^.  38.      8-3 

a^  —  1 


4-42^ 


(a  +  l)(a  +  2)(a  +  3)      ^     2  2^+17^-19 
•    (a  +  2)(a4-3)(a  +  4)  *      Z:^-bx^-1 


4^:2^3^,22  ^2     ^-(y-gy 

"    b^-Zx-X^  '    (x  +  yy-z^ 


43. 


FRACTIONS  161 

(x  +  zy-y^  '  *    x^-a^  +  yi^x-^-y) 


^g     7?-x^y-\-xy'^-7?  ^g     \-x-^y-xy 


a?  +  o^y  +  xy"^  +  ^  1—  x  —  z  +  xz 

a^-\-2x^-\-x-{-2  4  52^  _  (52  ^  g2  _  ^2)2 

a5  +  3a;24.a,^3*  *    4c2a2_  (^.^  ^2.  52)2" 


Multiplication  of  Fractions 
132.   Multiplication  of  a  fraction  by  an  integral  expres- 
sion.    Let  ^   denote  any  fraction  and  c  any  integral  ex- 

pression.     Representing  the  quotient  of  -  by  5^  we  have, 

i=..  (1) 

Since  the  dividend  is  equal  to  the  product  of  the  divisor 

and  the  quotient, 

a  =  bq.  (2) 

Multiplying  both  members  of  identity  (2)  by  c, 

ac  =  bcq.  (3) 

Dividing  both  members  of  identity  (3)  by  6, 

ac      bcq  a  ^^ . 

T=   6   =^^  =  ^^ft  (4) 

That  is,  ^=^Xt-  (5) 

0  0 

.      .  ac      ac  -^  c         a  ^ci\ 

Again,  —  = = (6) 

b        0-7-  c      b-i-c 

Therefore,  from  identities  (5)  and  (6), 

=  cxl  (7) 


b-i-c  I? 


162  ELEMENTARY  ALGEBRA 

Identity  (5)  shows  that  multiplying  the  numerator  of 
-  by  <?  multiplies  -  by  c.  and  identity  (7)  shows  that  divid- 

0  0 

ing  the  denominator  of  -  by  <?  multiplies  r  by  <?;  hence, 

0  0 

Rule.  To  multiply  a  fraction  hy  an  integral  expression^ 
either  multiply  the  numerator  or  divide  the  denominator  hy 
that  expression. 

Note.     Divide  the  denominator  when  possible. 


ILLUSTRATIVE  EXAMPLES 

1.  Find  the  product  of  —  and  a. 

n 

Solution.  ax-  =  — •  '  [§132] 

n        n 

2.  Find  the  product  of  ^  and  h. 

0 

Solution.  bx^=  -^  =  ?  =  a.  [§  132] 

0       0  -^  0        1 


3.    Multiply  —  hy  4  ah, 
xy 

Solution.  —  X  4  a6  = 

xy  xy 


2    1       2 

4.    Multiply  ^^_^  by  a;  -  ?/. 


(a;8  -  y8)  -=-  (a:  -  y)      x^  +  xy  -\-  y^ 


Solution,      -^-^f^  '  (^  -  y)  = 


5.    Multiply  ^  by  a:"»+i. 

Solution.  — x  »»+i  = = 


FRACTIONS  163 

EXERCISE  64 

(Solve  as  many  as  possible  at  sight.) 
Multiply  as  indicated : 

1.    2x|.  2.    2xi.  3.    CX^' 

5  8  n 

*.p.l.  5.   2ax|.         e.   *x^. 

7.    2X       ^^      .         8.    <5<fx-g^.  9.       baxx   ^^^ 

10.    9a;y2X^^^.  11.    (a  +  i)xl. 

12.    (m-n)x—-^—-^  13.    (a  +  6)x^. 

6(7w  —  n)  c  —  a 

{x-yY  x^  —  y^ 

Multiply : 

-    !a^^^v(a-H5).  17.    5±|by8(.  +  *). 

IS-  J^o^  ^  hy^-x^X,       19.   -^,  byaf*. 

^'^  I  ''y  *•  "•  |;r  by  ^• 


23. 


g?jg7^'>^<'-»)<«-.0. 


24.     _?-^bv(a^  +  y)2.        25.     r^>y5. 


164  ELEMENTARY  ALGEBRA 

Addition  and  Subtraction  of  Fractions 

133.  Adding  and  subtracting  fractions  with  the   same 
denominator.     By  the  distributive  law,  Section  60, 

\m     m     mj         \mj         \mj         \mj 

=  a  +  6-|-c  [§  132,  Rule].         (1) 

Dividing  both  members  of  identity  (1)  by  m, 

«4.1  +  £  =  ^±M^.  (2) 

m     m     m  tn 

Again,        mi— ]=m[—]—m[  —  \ 

\m     mJ        \mj        \mj 

=  a-b.  (3) 


Dividing  both  members  of  identity  (3)  by  w, 
a      b     a—  b 


mm        m 


(4) 


From  identities  (2)  and  (4),  we  have  the  following 
Rule.      To  add  or  subtract  fractions  which  have  the  same 

denominator^  add  or  subtract  their  numerators  and  place  the 

result  over  their  common  denominator. 

134.  Lowest  common  denominator.  Two  or  more  frac- 
tions whose  denominators  are  not  the  same  may  be  re- 
placed by  other  fractions  equivalent  to  them,  respectively, 
each  of  whose  denominators  is  the  lowest  common  multiple 
of  the  denominators  of  the  given  fractions. 

The  lowest  common  multiple  of  the  denominators  of 
two  or  more  fractions  is  called  their  lowest  common 
denominator  (L.  C.  D.). 


FRACTIONS  165 


ILLUSTRATIVE  EXAMPLES 

1.  Express  as  a  single  fraction  --\-- • 

a      b      e 

Solution.     The  L.  C.  D.  of  the  fractions  is  abc. 

a     abc^ 
1  _  ac 
b     dbc^ 

c         abc 

abc 

2.  Express  as  a  single  fraction  in  lowest  terms : 


x^  +  x     a^+Sx-t2      x^^-lx 
Solution.     The  L.  C.  D.  of  the  fractions  is  x{x  +  l)(x  +  2). 

1  a:  +  2 


x^ -\- X      x(a:+ l)(a:+ 2) 
1  X 


a;2  +  3a;+2      x{x  ^  l)(a:  +  2) 
1  2;  +  ! 


a:2  +  2x  a:(a:  +  l)(a:  +  2) 


Sum      =(^  +  ^)+^-(^  +  ^> 
a:(a:+l)(a:  +  2) 

_a;  +  2  +  a:-a;-l 

a;(a:  +  l)(a:4-2) 

1 

a:(x  +  2)  * 

3.    Express  as  a  single  fraction  ^  ■\- xy  -\- y^  ■{ — ^ — 

x-y 

Solution.     :r2  +  zy  +  y2  +  _J^  =  £iHl£y±l!  +  -J^ 
a;  -y  1  x-y 

a:-  y       a:  -  y 
a:8 


166  ELEMENTARY  ALGEBRA 

4.    Simplify: 

b  -{-  c  ,  c-^  a  a-\-h 


(^a  —  b){a  —  c}      (h  —  c)(h  —  a)      (^c  —  a)(c  —  b^ 
Solution.     ^-±^ + i±i5 +         «  +  ^ 


(^a-b)(a-c)      (b-c)(b-a)      (r  -  a)(c  -  b) 
b  -\-  c  c  +  a  a  -\-  b 


(a-bXc-a)      (a-b)(b-c)      (b  -  c){c  -  a) 

The  L.  C.  D.  of  the  fractions  is  (b  —  c){c  —  a)  (a  —  b). 
b  +  c  _  (6  +  c)(b  -  c) 

(a  —  b)(a  —  c)  (b  —  c)(c  —  a)(a  —  b) ' 

c  +  a (c  -\-  a)(c  —  a) 

(b  -  c){b  -a)~      (b-  c)(c  -  a){a  -  b)' 

a  +  b         _  (a-\-  b)(a  -  b) 

(c  -a)(c  -  b)  (b  -  c)(c  -  a)(a  -  b)' 

(b  —  c)(c  —  a)  (a  —  b) 
_  52  _f.  c2  _  c2  +  a2  -  a2  _,.  52 


(b  —  c)(c  —  a)  (a  —  6) 


0. 


From  the  solutions  of  illustrative  examples  1,  2,  3,  and 
4,  pages  165  and  166,  we  have  the  following: 
Rule.     For  adding  or  subtracting  fractions : 

1 .  Reduce^  when  necessary^  the  fractions  to  their  lowest 
common  denominator. 

2.  Find  the  algebraic  sum  of  the  numerators  of  the  re- 
suiting  fractions. 

3.  Write  the  algebraic  sum  of  the  numerators  for  the  nu- 
merator of  the  result  and  the  lowest  common  multiple  of  the 
denominators  for  its  denominator. 

4.  Simplify  the  resulting  fraction. 

Note.  In  general,  before  operations  on  fractions  are  performed, 
each  fraction  involved  should  be  reduced  to  its  lowest  terms. 


FRACTIONS  167 


EXERCISE  65 


Express  as  a  single  fraction  in  lowest  terms : 

1.     l  +  l-  2.     1+1.     3.    ^  +  ^.  4.     2x  +  ^ 

a     0                             a           zx      Sx  x 

S.    x+±-f.  6.     a  +  *.    7.     X-&.  8.     1-1 

ZX     OX                  c                   z  he      ca 

4:a      6a  a+la 

13.     -f+*.  X4.         1  1 


ha  a-\-  h     a  —  h 

15.     -i-  +  l.  16.     _2--A. 

2H-a     a  a  +  2      2a 

17.     -^  +  .  ^     ■  18.         '  2 


x  +  ^z      2a:+2y                      a:  +  3      a;+2 
19. — •  20. + 


aH-2      3a-l  a-1     a  +  1 

21.     -  +  -.  22.     ^-hf-' 
z      X  by      OX 

x^  1  1 

23.     2: 24. 


a;  +  ^  2a  +  6      3a  +  J 

25.     ^  +  ?^.  26.         1      +-J_. 

22;  2y  a-\-h      c  +  d 

27.    1+1.  28.    9_t2a^9-25 


a-1  3a  36 


168  ELEMENTARY  ALGEBRA 


31.     —±-,^^-±^.      32.         ^ 


33. 


(a;+y)5      {x+i/}a  a^  —  ab      P  —  ab 

la     /a-\-5b      b—2a\ 


-( 


b   \  b  ■     b  J 

34.     B-^  +  £.  35.     f'  +  *i+4- 

ax     ox     ex  DC     ca     ab 

36.     ^±l-l±l^^±l.  37.    ?_^4.i5 

a:z/          2/5J          za:  a      3a      9a 

38. -•  39 


2a(a;-a)     2a(a;+a)  6(2  a;- 3)      6(2  a: +  3) 

40.    — - — I 41.    a  H-  6 


a^        a*  a  +  5 

42.     1 •  43. 


a      a+1     •  Sx^—Sxy      '^iy'^  —  ^yx 

o^  _i_  ^  _  b^i^^-^y^)  _  3  a^^ 

45.       1     +-J_+     1     .      46.    ^+     ^ 


6  — c      c— a      a  — 6  b  —  c     c  —  a      a  —  b 


49-     -^ T^T r  + 


50 


(a  -  6)(a  4-  <?)      (b  —  a)(b  +  <?) 
1  3 


(a  -  6)8      (5  -  a)3 

gj^^    3^-\-xy-\-y^     x^-xy-\-y^ 
a^  —  y^  a^  +  y^ 

52.  2a;y     ^2x-{-y     x-\-y 
x^-y^      x->ty      y  —  x 

53.  1+^  +  ^   '        ^  +  ^ 


a;a;2_9      x^-^x-\-^ 


FRACTIONS  169 


54  flt  +  g  .  h4-c 

55.     ..         il  4  2  .  8 


56. 


(3a:+2X2a;-3)      (22:-3Xa:-2)     (2-2:X3a:+2) 

2 4 5_ 

2^24.43.4.3      a,2_2a;-3      l-a:2* 


57.     Z § + I 

2a^-bx--Z      ^a^-lOx^Z      Q:^^x-1 

x-^\  942:- 186       .       71a:-135 

08.    — — — — — —  —  — — — - — — — —  4" 


Qx^-llx+12      152^-14a:-8      Qx^-Vlx+ll  . 

In  examples  59-64,  combine  not  more  than  two  fractions 
at  a  time.  Thus,  in  example  59,  combine  the  first  two 
fractions  and  then  the  result  with  the  third. 

59.  ^+-^+  2 


1  —  a      1  +  a      l  +  a^ 

a  h  -2_A! 

60.     -^  +  —^4 


a  +  h      a-h      a^-^h^ 

x  —  y     x-^y     a^  +  y^     x^ -\- y^ 

__         a         a4-4,      a         a  —  4 

oz. \- • 

a +  4         a         a  —  4         a 

a^+x-h^      a^-1  2a;-8 


63. 


x-1  x-2      x^-Sx  +  2 


64.     -^+      1  1 


x-S     x-\-S     x-2     x  +  2 
gg      3(22;2+l)         2a:+l  2a:-l 

3^^x^+l      x^-x-^1      x^^-x^-1 

Reduction  of  Fractions  to  Integral  or  Mixed  Expressions 

135.   Simple  fraction.      When   both   terms  of   a   frac- 
tion are  integral,  the  fraction  is  called  a  simple  fraction. 


170  ELEMENTARY  ALGEBRA 

136.  Proper  and  improper  fractions.  Simple  fractions 
are  classified  as  proper  fractions  and  improper  fractions. 
A  proper  fraction  is  a  simple  fraction  in  which  the  degree 
of  the  numerator  is  less  than  the  degree  of  the  denomi- 
nator. An  improper  fraction  is  a  simple  fraction  in  which 
the  degree  of  the  numerator  is  either  equal  to  or  greater 
than  the  degree  of  the  denominator. 

Thus,  — is  a  proper  fraction ;  —  is  an  improper  fraction. 

a^  +  o  a  -\-  h^ 

Remark.     When  the  terms  of  a  fraction  contain  more  than  one 

literal  number,  the  fraction  may  be  a  proper  fraction  with  respect  to 

one   of   these  numbers  and  an  improper  fraction  with  respect  to 

another. 

Thus,  — — —  is  a  proper  fraction  with  respect  to  6,  but  it  is  an 
a  -\-  h^ 

improper  fraction  with  respect  to  a. 

137.  Mixed  expression.  An  expression,  some  of  whose 
terms  are  integral  and  some  fractional,  is  called  a  mixed 
expression. 

Thus,  a  +  -,  X  -\-  y  — ,  and  1  +  -  are  mixed  expressions. 

c  ^       x-\-y  y 

Note.  An  improper  fraction  can  be  reduced  either  to  an  integral 
expression  or  to  a  mixed  expression  in  which  the  fractional  part  is  a 
proper  fraction. 

ILLUSTRATIVE   EXAMPLES 

1.    Reduce  r to  a  mixed  expression. 


Solution  1. 


a2  _  52  _  q  _!■  2  h     a^-b'^-a  +  b  +  b 
a  —  b 


a-b 

a2 

-62 

-b 

—  a 

:  +  b 

h 

a 

a  - 

-b    -^a 

— 

b 

a 

-62 
-b 

a  — 
a  — 

b         b 
b^a- 

1 

a 

+  6- 

-1  + 

h 
I, 

FRACTIONS  171 

Remark.     Since =  —  +  —  —  —    it  is  evident  that  a  f rac- 

m  m     m     m 

tion  whose  numerator  is  a  polynomial  can  always  be  written  as  the 

algebraic  sum  of  two  or  more  fractions.     As  a  step  in   reducing   a 

fraction  to  a  mixed  expression,  it  is  desirable  to  express  it  in  this 

manner,  whenever  the  terms  can  be  grouped  at  sight   in  such  a  way 

that  each  numerator  with,  in  general,  the  exception  of  the  last,  is 

exactly  divisible  by  the  denominator. 


Solution  2. 

a  - 

a-l  +  & 

_  j)a2  _  a            _  ^,2  ^  2  6 
a^          —  ah 

-a  +  ab-b^-{-2b 

-a                   +6 

ab-b^  + 
ab-b^ 


a^-a-b^  +  2b 

=  a  —  1  +  0  + 


a-b 


Remark.     Solution  2  is  preferable  when  it   is  not   evident   at 
sight  how  the  terms  of  the  numerator  should  be  grouped. 

Check.     Let  «  =  2,  6  =  1. 

Dividend  =  Quotient  x  Divisor  +  Remainder. 
ai-b^-a-\-2b=  (a-{-b-l)(a-b)  -{-b. 
4_1  _2  +  2=(2+l-l)(2-l)  +  l 
3=2x1+1 
3  =  3. 

2.    Reduce  2a:«  -  3a:V  +  4r.y^  + 5y3  ^^  ^  ^.^^^  ^^ 
S  x^ -\- 2  XT/  —  4:  y^ 


sion. 


X  -  -U 


til. 


Solution.     Sx^  -\-2xy  -4:  y^)2  x^  -  S  x^y  -\- 4:  xy^  +  6  y^ 

2x^  -^  ^x^y  -fxyg 

-^x^y  +  \^-xy^+5y^ 


172  ELEMENTARY  ALGEBRA 

Hence, 

3  a:2  +  2  xy  -  4  2/2  3  9  ^      3  x^  +  2  xy  -  ^  y^ 

=  ^x-^y+ S6xy^-7y^ 

3  9   "^      9(3x^  +  2xy-iy2) 

Check :  Let  x  =  2,  y  =  1. 

2aH»-3x2.y  +  4xyH52/^==(|x-^3,)(3x2  +  2x3/-4y2)  +  ?^^^^^ 

16  _  12  4-  8  +  5  =  (i-MVl2  +  4  -  4)+^^^~T. 
\3      9  /  ^  ^  9 

17=-f  +  ^^  =  17. 

EXERCISE  66 

(Solve  as  many  as  possible  at  sight.) 
Reduce  to  either  an  integral  or  a  mixed  expression  : 


1. 
3. 
5. 
7. 


a 

2a^ 

-  3  rr2  +  2  a;  - 

1 

X 

4  w 

-f-  3  mn  —  2m 

n 

x^- 

-a2 

X  — 

■a 

x^- 

-2ir  +  l 

x-1 

a2_ 

b^  +  a~b 

a-b 

2a2 

-ft2  +  l 

mn 

Sa^-2ab-\-5       ' 

2a 
l  +  3a6-2a2j  +  5j8 


11. 


a-\-b 
15.     ^-y'. 

X+1/ 


2ab 

8. 

ma-\-  mb 
a  +  b 

10 

2  'irRH+  2  irm 

H+R 

12. 

a^+b^ 
a-b 

14 

a^-b^j^a-2b 

a-b 

16 

2x'^-\-^xy  +  y^ 

2x^Zy 

FRACTIONS 


173 


17. 


19. 


21. 


3  a;2  -  2  rg,y  +  2 
3a;-2«^ 

x^-\-2xy  ■\-'^y'^ 
Sx^-2x-\-2 
2a^-^x-\-S' 


18. 


20. 


22. 


Sx^-\-Sxy  +  2 
x  +  y 

2a^+3a;+l 

x^+2x-,^  ' 

Zx^-2xy  +  4:y^ 
2x^-^4xy-Sy^' 


Multiplication  of  Fractions 

138.    Product  of  two  fractions. 

Let  j=:q       (1)     and     ^  =  r 

0  a 

Then,  a=  bq     (3)     and     c  =  dr 

ac  =  hdqr. 

Dividing  both  members  of  (5)  by  hd^ 

ac 


(2) 

(4) 
(5) 


[§  25,  3] 


bd 


=  qr. 


But 


a     c 

H  v£  — ^ 
b     d~bd' 


(6) 

a) 

(8) 


From  identity  (8)  we  have  the  following  : 

Rule.      To  find  the  product  of  two  fractions,  multiply  the 

numerators  together  for  the  numerator  of  the  product  and 

the  denominators  for  the  denominator. 

ILLUSTRATIVE    EXAMPLES 

Scd 


1.    Find  the  product  of  ^ — i::  and 


Scd^ 


^ah^ 


Solution. 


2a% 
3c(P 


Scd 


a 

2hd 


(I 
2M 


Remark.     In  practice  it  is  customary  to  cancel  as  shown  in  solu 
tion  of  example  2,  which  follows. 


''^'^  :,2  +  5^+6 

"^  4 

a:-4 

x^-hSx-4: 

3a: 

-k6_ 

=  (3>^)(^  + 

4) 

3(^^K2) 

'     a;2+5a;  +  6 

4a; 

-4 

(2>K2)(a:  + 
3(a:  +  4) 
4(a:  +  3) 

3) 

i(^^^) 

Let  x  =  2. 

a;2  +  3  a;  _  4 

3a; 

+  6 

=  3(^  +  4). 

a:2+5a;  +  6 

4a; 

-4 

4(a;  +  3) 

4  +  6-4 

6  +  6. 

_3(2  +  4). 

174  ELEMENTARY  ALGEBRA 

2.    Multiply  V.,.         • 

Solution. 

Check. 


4  +  10  +  6      8-4      4(2  +  3) 

Remark.  It  is  evident  that  x  —  1  and  a;  +  2  may  each  be  can- 
celled in  the  numerator  of  the  one  fraction  and  the  denominator  of 
the  other  [Solution,  example  2],  for,  if  expressed  in  the  product,  each 
would  be  a  factor  common  to  the  numerator  and  denominator  of  the 
product  and  hence  could  be  cancelled. 

3.    S>mphfy  1  -  ^,  .  ^^^/^^  •  Vzfr- 
Solution. 

(C  _  y)2    '    c*  +  CV  ^  y4         c^  _  yi 

^l      (c  +  y)(c^-cy  +  y^)  ^       (c  -  y)(c^  -[-  cy  -{-  y^)       ^        (c  +  .y)2 

(c  -  2^)  2  (c2-  cy  +  y^)(c2+  C3/  +  y2)      (c  -  2/) (c  +  y) 

^1     (^  +  yy 
{c-yy 

^(c-yy-(c  +  yy 
(c  -  yy 

^l(c-y)  +  (c  +  y)M(c  -  y)  -  (c  +  y)] 
^(2c)(-2y)^   -4:cy 

{c  -  yy     (c  -  yy 


FRACTIONS  175 

EXERCISE   67 

(Solve  as  many  as  possible  at  sight.) 
Simplify : 

2a  2^—  si—  4^§f 

3*5*  '    Sh'b^'  '    a^'a^'  '     b   '  d' 

5.    ^.?5.J_.  e.    2a. ^. 

5      21    -  2  c 

„      2    «^  o     2a   4  5 

2xy^   4x  —  3 mn^p^   4m 

*  3a52*3z/'  '       2a5V     *9a* 
5a:yV    -2a;y  2  a^^gS    4  a^^g^ 

'      3a5g  *     3ag    '  *    Sde^^' 9  cPef^' 

2(x  +  ^)    (x  +  yy  ^^     -3(a  +  5)      5(a -f  5)3 

3(2:-^)'(a;-^)3'  '     2(a-5)      -15(a-5)2* 

15.    ^  +  y    ^-y  16.    3(^H-^^^-^^ 

'    X  -  1/   2^ -\- 1/^  ^  —  y^x-\-y 

17     ^!zil    (^  +  ^)^  18     ^^    2(5+3) 

•  a2_52-     ^4.1  '54-3        3a2 


19. 


21. 


a52    6(^2    gf  4a5gc2    Ix^fz^     %ac 

—  . .  -sL  •  20.     . ^ • • 

cd^    e^f    ah  1  xyz     ZaVycH    %xyz 

a52/l_l\  _  20252/02  _a\ 

3U      hj  '            5     W      5/ 


23         a;         l-a2  2^     a2^2a  +  l    -352 

a  —  12x-\-xy  5^  o  +  l 


27. 


yv  z     \^ 

35-6a:    2a*+2a35     6x-^Sx^ 


2a  +  2x     dax-\-9x     2ab  —  ^ax 


176  ELEMENTARY  ALGEBRA 

28     <^^  -  2  aa;       ab-\-SP        2abi/-Sxy^ 
a^  +  Sab'  2abx-'d2^'i/'     ax-2a^ 

2m^-2     5?n2  +  5     bp^  +  bq^    bp  +  bq 
^p^-Sq'  ap^  +  aq^'  3^2+8'  cm-c' 

(y  +  l)2V4-l' 

b--a       a^-b^    6(x^-y^)    (x  +  y)2 
(a;  +  y)2'      10     *       a-^-b      '     x-y 


29. 


30. 


31. 


139.  Powers  of  a  fraction.  An  important  case  in  the 
multiplication  of  fractions  is  that  in  which  the  fractions 
to  be  multiplied  are  equal;  their  product  is,  therefore,  a 
power  of  one  of  the  given  equal  fractions. 


--     ^-hhh%=k 

m-^^^x=^>^^>^^-^- 

-32 
243 

32 
243 

(^y  =  ^^.^...  ton  factors 

_  a.  a.  a. ..ton  factors  ^a«j^^ 
b'b'b"-ton  factors       6'* 

W      IP 

0 

Identity  (1)  may  be  expressed  in  words  as  follows : 
Any  power  of  a  fraction  is  equal  to  the  same  power  of  the 
numerator  divided  by  the  same  power  of  the  denominator. 
Conversely,  we  may  write, 

?-©■•  « 


FRACTIONS  177 


ILLUSTRATIVE  EXAMPLES 
.2 


^'-"(ffDHfcf)' 


1. 

y 

Solution. 


\x  —  y  }      \ a  —  h  I        L         X  —  y        JL  a  —  6  J 

2.    Simplify  L________L. 

Solution.      (2  ^^  +  3. +  1)3^/2.^  +  3. +  1\» 

(X  +  1)8  V  X  +  1  I 

EXERCISE  68 

(Solve  as  many  as  possible  at  sight.) 
Raise  the  following  to  the  indicated  powers : 

^-  (2^)  •  ''  [-^r)       ''  (32^)  • 


10 


13 


IV 


3  xyH   J  \b  xy 

■  {-IT      '■  6)"- 

.  (ff.  „.  (=^)\    „.  (i+i) 

\oJ  V   xyz    )  \x     yj 

Suggestion.     Add  the  fractions  before  squaring. 

•e-p"  "S-'T-  -c--")" 

16  fi-iY.  17     i^'+'^y    18  C^'+^^'^' 

V        a:y  *    (a2-a+-l)3'        *     (a3  +  jsy 

19^  i^  -  ^yY  20.  i^tuMt..       21.    (^^  -  ^^^)^ . 

(^  —  a;)^  *      (a;  —  ^)^  *       (m^  —  w^)^ 


178  ELEMENTARY  ALGEBRA 

Write  each  of  the  following  as  the  square  of  a  fraction : 

•  J2_45_^4  •     a:2  +  2a;+l  "        *    / 

25     ^^  26       ^^  ^'^'^'  27      4-^-12a:+9 

*  9a456  *    64ar*2/V0*  *    9a:2^_i2:i;+4- 

Division  of  Fractions 

140.  Reciprocal  of  a  number.  The  reciprocal  of  a 
number  is  1  divided  by  that  number.  Hence,  whenever 
the  product  of  two  numbers  is  equal  to  1,  either  one  of 
these  numbers  is  the  reciprocal  of  the  other. 

Thus,  since  -  x  -  =  1,  the  reciprocal  of  -  is   -,  or  -  =  1  -f-  -  and 
ha  h         a         h  a 

-  =  1  -^-  -.     The  reciprocal  of  a  fraction  is  obtained  by  interchanging 
a  b 

the  numerator  and  the  denominator  of  the  fraction ;  that  is,  by  invert- 
ing the  fraction. 

141.  Quotient  of  two  fractions.  The  quotient  of  two 
fractions  may  be  obtained  by  use  of  the  identity 

Dividend  =  Divisor  x  Quotient  (V) 

Thus,  let  it  be  required  to  find  the  quotient  of  -  h-  -. 

0      a 

Multiplying  both  terms  of  -  by  cd, 

a  _  acd 
b      bed 

Expressing  the  second  member  of  (2)  as  the  product  of  two 
fractions, 

?  =  £  X  —  (3) 

b      d      be 

Observing  that    -    in  identity  (3)   corresponds    to   dividend    in 
b 

identity  (1),  that  -  in  identity  (3)  corresponds  to  divisor  in  identity 

(1),  and  that  —    corresponds  to  quotient  in  identity  (1),  and  since 
be 


(2) 


FRACTIONS  179 

dividend  divided  by  divisor  is  equal  to  quotient,  we  have, 

a  ^  c  _  ad  _  a      d  ft^. 

b      d      be      b      c 

From  identity  (4)  we  have  the  following: 
Rule.      To  find  the  quotient  of  two  fractions  multiple/  the 
dividend  hy  the  reciprocal  of  the  divisor, 

142.  Two  special  cases  of  division. 

1.  When  the  dividend  is  a  fraction  and  the  divisor  is 
an  integral  expression. 

Any  integral  expression  can  be  written  in  the  fractional 
form. 

Thus,  c  may  be  written  -. 

Therefore,  --c=-h--  =  -x-  =  -.  (1) 

b  b     1      b      c      be 

That  is,  «^c=-^.  (2) 

b  be 

From  identities  (2)  and  (3) 

From  identities  (2)  and  (4)  we  have  the  following: 
Rule.      To   divide  a  fraction  hy  an   integral   expression^ 

either  divide  the  numerator  or  multiply  the  denominator  of 

the  fraction  hy  the  integral  expression. 

Note.  In  practice,  divide  the  numerator  rather  than  multiply  the 
denominator  whenever  the  numerator  is  exactly  divisible  by  the 
integral  expression. 

2.  When  the  dividend  is  an  integral  expression  and 
the  divisor  is  a  fraction. 

a^^  =  «^^  =  «x^  =  ^.  (1) 

c     \     c     1     b      b  ^  ^ 

That  is,  a-^  =  ^.  (2) 

c      b 


180  ELEMENTARY  ALGEBRA 

From  identity  (2)  we  have  the  following: 

Rule.      To  divide  an  integral  expression  by  a  fraction^  find 

the  product  of  the  reciprocal  of  the  fraction  and  the  integral 

expression. 

ILLUSTRATIVE  EXAMPLES 

1.    Divide  — —  by  — —. 

21  xy^^  28  xy^ 

2ab  4 

Solution       IQ  ""^^^  ^  -  15  ab^c^  _  ^0^^^^    .2S-^vt^ 
21  xy^z^  '     28  xyz»        -iirxy^z^    ^-l^^*3c»- 
Sy  -3c2 

Sy  3c2 

__  Sab 

Solution.      £^x3£%^^=^x-i:^x:i^*t' 

«        2  J 

a       2        ^ 

2  ay' 

q.      ,.£       (rg  H-  y^^  x^  -\-  xy         .  (a^  +  y)^ 

x^  —  y^       (x  -\-  ^)2  —  xy       7^  —  y^' 
Solution. 

(a^  +  y)^  j^       a:^  +  xy       ^  {x  +  yY 

=  7^7^  — ^^^ 

1     1     x  +  y 

__      X 
~  x-\-y 


FRACTIONS  181 

EXERCISE  69 

(Solve  as  many  as  possible  at  sight.) 
Simplify : 

1.    — '- m.  2.    —-5-6.  3.    --5-C. 

n  0  0 

1                     -             w  ^  ^ 

4.     a-i--.  5.    m-5 .  6.     r-=--. 

c  7i  8 

c  c 

9.    _^?_^(a;+i/).  10.    '^^-^^^(m-n^ 

X  —  y  X 

U.    "''  +  ^'^(,»  +  n).  12.     ^^(x-1). 

13.      (a:-3/)-5-^— .  14.       (2^_^2)^£l^. 

a^  +  y  a 

a  x-\-  Z 

a  0 

19.     -^ r-^  -5-  «(6  -h  c).        20.     — 5-  (w  +  ri). 

0  —  1  m 

«,         1         1  oo        3        4  o,        1         1 

21.     --5--.  22.     --5--.  23.     -^-. 

X      2  a      0  X     y 

«.      «     2                 ^^     2a3     4a2         ^^3         5 
24.     — ; — .  25. ! .         26. ; . 

2      a  3  63     9  63  2a     4a2 

27.     ^-^^.  28.     ^-5-^1^. 

63  3  62  I  y 

5£^_15^  3^      (g-f ^>)^    .  2Ca-h6)8 

Zmy^'   2m^f'  *     3(a:-2/)S  *  5(a:-^)2' 


182  ELEMENTARY  ALGEBRA 


31. 

2ab,Zb           32 

a 

2d 
Zcx 

33. 

2a  .  8a 
36  *  3  6* 

34. 

^-^.               35. 
cd  '  bd 

ax  ^  X 

%  ■  y' 

36. 

2a6     2  6 
Zed'  Zc 

37. 

2  a5V      3  a^^c 
9  xi/^^     4  3^y^z 

38. 

H-- 

39. 

-■... 

40. 

41. 

.-^.!|?. 

42. 

(«+6)2h 

2(a  +  6)3 
3a 

,^      -2a2:        2x                    _       -a362c4      2  a36c2 
^^^     _t_ ^  44. ! • 

3  y2     *   _  cfiy'  '      2  a;2«/32    *  3  x^y'^z 

a%      b'^c      abx  ^^      .        -,x      x^-\-x-\-l 

45.     -^-^—7r~- •  *^-     (^  — 1)"^ T"^ • 

x^y     y^z      yz  1  —  x 

47.      (2;2_.  y2)^^Zll.  48.      (2:  +  a)2H--^±^. 

^!±i!^(«2  +  ,j  +  52).       50.     2(:r+y)^^3(.+  y)8 
a-6      ^  ^     ^  .  3(:r-^)3      2(x-yy 

(2a;+l)(3a;-2)  .  (3  rr- 2)(2  a;  + 1)2 
(3a:+2)(2a:-l)  *  (3  2:  +  2)(2  2:  -  1)2* 

(2a;+3)(2  2;-3)  .    (2  a;  + 5)(2  a;- 3) 
(3a:  +  2)(3a:-l)   *  3(3a:-l)(2a:-l)* 

6a;2H-5a;-6  .  2a;  +  3 
Qx'^-bx-Q^  Zx  +  2' 

9^:2-1  3  2:2 +  2  a;- 1 


49. 


51. 


52. 


54. 


6  2;2-5a;  +  6*    2a:2-2;-3' 

f  -  x^y   ^  fy  -  a^Y 
-\-  yx-\-x^      \y  ■\-  x) 


y^^-y)     y'^ 

a^  -\-  ab  —  ae      (a  —  c^  —  b^  a 


57. 


58. 


59. 


FRACTIONS  183 

3  x^y^  +  3  +  6  a:!/  ,  "Ixy-^-l 
4x^7/^ +  4: -8x1/  '  Sxy-S' 

2^3  _  y3  x^  —  xy  +  y^  _^  x^  -{-  xy  -^  xz-\- yz 

x^  —  xy  -\-  xz  —  yz  x-\-  y  a^ -j-  x^y^  +  y^ 

(x-^yy-z^  ^  z^-x'^-\-yiy-2z)   ,  (y  +  z^-x^ 
^x-yy-z^      z^-x^  +  y(2x-y}  *  (y-z^-x^' 


60.  What  is  the  reciprocal  of  —  f  ?     Of  —  -  ? 

61.  What  is  the  reciprocal  of  the  reciprocal  of  a  frac- 
tion ?     Illustrate  by  taking  ^. 

62.  When  equal  factors  are  cancelled  from  the  numera- 
tor and  the  denominator  of  a  fraction,  what  operation  are 
we  performing  on  the  terms  of  the  fraction  ? 

63.  Why  can  we  not  cancel  the  like  terms  in  the  nu- 

2  -\-x  .2  +  2^     2  +  1 

merator  and  denominator  of  _         and  obtain = - 

o  1+x  1+/     1+1 

=  -9 

2' 

X  A-  \ 

64.  For  what  value  of  x  has  no  meaning? 


Complex  Fractions 

143.  Definition.  A  complex  fraction  is  a  fraction  which 
has  one  or  more  fractions  in  either  or  both  of  its  terms.  A 
complex  fraction  is  said  to  be  simplified  when  it  is  reduced 
to  an  equivalent  simple  fraction  or  integral  expression. 

In  simplifying  a  complex  fraction  it  is  usually  most  convenient  to 
express  each  term  of  the  fraction  in  its  simplest  form  before  attempt- 
ing to  perform  the  indicated  division.  Sometimes,  however,  labor 
is  saved  by  first  multiplying  both  numerator  and  denominator  by 
the  L.  C.  D.  of  aU  the  fractions  contained  in  the  terms  of  the  given 
fraction.  . 


184  ELEMENTARY  ALGEBRA 


ILLUSTRATIVE   EXAMPLES 


1  +  1 

1.    Simplify    =-  - 


Solution.     Multiplying  both  terms  of  the  given  fraction  by  ar, 

x_ X  +  1 


1 


2.    Simplify    :j T-'^'i T"* 

a     b-\-  c     b      a-\-  c 

Solution.     Multiplying    both    terms   of    the    first    fraction    by 
a(h  +  c)  and  both  terms  of  the  second  fraction  by  b(a  4-  c),  we  have, 

1 1_     1 1_ 

a     b  -\-  c      b      a  -\-  c      b  +  c  —  a      a  +  c  —  h 


i+ 

a 

1 

6  +  c 

r 

b 

^     1          b  +  c  -\-a 

a  +  c 

_b  -\-  c  -  a 

a+  c 

+  b 
+  6 

b-\-c  +  a 

a-\-c 

-b 

b  +  c  -  a 

a  -\-  c  —  b 

EXERCISE  70 

Simplify  : 

'^- 

-=t- 

3. 

-2.5 

-^ 

a 

4.1 
c 

b 

mn 
5     ""^^ 
ab 

6. 

m  +  Si 
.        1 

1+^ 


FRACTIONS  185 

^  —  yi- 


^  _mr  ^—  y 

n^  a 

4__  X 

X 


10. 11. 

a 

1 


is  _0'  X  —  a 


X 


X—  y     X  —  z 
13.    ^ 14, 

V  z 


r-1 

9. 

r+l 
r+1 

r-1 

X       w 

12. 

y    z 

X      w 

y    z 

1       , 

1 

a  +  1  '   a 

;-i 

1 

1 

y—x     z—x  a—\      a+1 

(j-')^')Hr')e-') 


15. 

1+^ 


62 


16.    T— 70 17.    j—-^ 

ao  +  tr  a—  0  -\-  c 

ab—l^  a—h—c 

X     y     fl   .  1\2  1^1         1^1 


X      \x      yj 


y      X      \x      yj  x     y  -\-  z      X      z  —  y 

18. ! — = = —  •  19.    = = '-  T — . 

x     y       1^_J^  1         1         1         1 

y     X      a^     y^  X     y  —  z     X     y -\- z 
22. 


186  ELEMENTARY  ALGEBRA 

(i — ^ — —V(— — —)' 

\        i/-\-z     x  +  yj     \y  +  z     x-Vyl 


23 

V       \ 

1 


24.   x-\- 


.+1 

X 


Suggestion.  Multiply  both  terms  of  the  complex  fraction  by  x 
and  then  reduce  the  resulting  mixed  number  to  an  improper  fraction. 
A  complex  fraction  of  the  form  given  in  this  example  is  called 
a  continued  fraction. 

1  3 

25.    iH 7.  26 


1+i*  '    24-      ^ 


oc^  —  y^ 


X  2-\-x 

EXERCISE  71.— REVIEW 

1.  Name  at  sight  the  result  in  each  of  the  following  : 

b  X     m^  f       Tt 

ax-;  ax—; ^-m;  -z=i-i-  rt. 

c  ae     n  R 

2.  Name  at  sight  the  result  in  each  of  the  following  : 

^{x-y^, 

z 

3.  Reduce  — — — -  to  lowest  terms. 

arm  +  3  a^  -f  am  +  3  a 

4.  Does("-y=-(^^l^%     Why? 

c  —  a  d  —  c 

( 'YY\.  —  92,  I  ^ 

5.  Reduce  1  —  ^^— ^  to  an  improper  fraction. 

^2  ^  ^2  ^     ^ 

\  —  ^ 

6.  Reduce to  a  mixed  expression. 

7.  Add  -  to  a. 


FRACTIONS  187 


8.    Subtract  x  from  - 


Si„pli„g.f).g-f). 


Simplify  : 
1 


10 


m-\-n     m—  n 


11.  J-+-1-, 

12.  1 


2  X— y      Z  x  —  ly 


13.  ^^^ 


I  ^% 


nhx  -f  ^i^y      max  -\-  may 
14.    ,      ^  .      +  „      1   .      +       1-6^ 


15.   -JL+     2     ^       J_  3a^ 


a  —  5      a  -f-  5      a^  —  b^      (a  —  by 
16.    ^  +  .-^„+       1 


a;— 1      1  —  a;2      a:^  —  1 
17.    -1-+      1  1  1 


x-\-a      x—a      x  —  b      x-\-b 

^^y     y ~~ ^     z  —  x     x  —  z     y  —  x     z  —  y 


a  +  36      a-36      a3_9^52 
20.    fl-    /-g      Y-±l. 

\  3?-X-Vx-\ 

■i-a  l-a2 


188  ELEMENTARY  ALGEBRA 

.1.1      a3  +  3a2_l 

22.    aH-l+ \vo 

a  a^  -{-2  a 

ix  +  2yy      ix-2yy 

24.    l-^_  +  -J_  + ^+1 

a     a— 1      a  — 2      a(a  — l)(a  — 2) 


26. 


1_1 ■ 1^1 

X     y      X     y 


Suggestion.     Factor  the  resulting  numerator  by  arranging  accord- 
ing to  powers  of  a. 


(jd  —  h)(a—  c)      (h  —  c)(h  —  a)      (^c  —  d)(^c  —  b) 
28.       ,         i,  ,4-  1  1 


a(a— 6)(a— (?)       h(h  —  d)(h  —  c)       e(^c  —  d)(^c—  h) 


29.    a  +  a; 
30. 


a—  a;        a  -\-x 
1  1 


(:r4-l)(^H-2)       (a;  +  2)(a:  +  3)       (a;4- 3)(2:  + 1) 
1  a^-1  0^4-1 


34.    - — 7  + 


x^-l'x^  +  1      a^  +  a^  +  1      2^-a^  +  l 


FRACTIONS  189 

•       ^   x+1   ^  (a: +  3)2^  (a; +  3)2(2^+1) 
2a^  10  4 


36.    x  +  S-^ 


(a; -1)2      2;2_i      (a;+ l)(a;- 1)2 

x-^1 

37  (2xy  +  ^x-y      2xy-^x-y\  .        a^y 

V      32;-^  32:  +  y       J  '  ^x^-f 

a^  +  2a;+3      ^2^2^  +  3 
X  y 

38  -^ 


3a^  +  2a;+l      3^/2+2^  +  1 


rc2  4-  a:  a^^+7a;  +  10      x^  +  1  x-^12 


a^  +  5a:  +  4  a:2^2a;  2^  +  82;  + 15 


1      r^      I2x-1      lfx-2      (a;  +  l)(a:-3)\n 
*    a;     L        1       ^  3U  +  2  a^(a;  +  2)      jlj 


41 

X 

3  8A2  +  7c?2  2^-6<? 


42 


43. 


2A  +  (?       8^3+^3      4A2-2Ac  +  c2 

3a;-l      2a:  +  3      5a:2_2a;-l 
a;+3         1-a:         ar2  +  2a;-3  ' 


44  <?  +  ^      ^2  _,_  ^5  ^  ^2 

a-6      "*"      a2  _  52      • 

a^  —  y^  x^-\-  y^ 

a^-b^               '^^^^           a2_^ 
45.     X  X  

0^ -\- xy  -\-  y'^      x^  —  xy-\-  y'^     x^  —  y^ 

a  —  b  a-{-b 


190  ELEMENTARY  ALGEBRA 


a^  +  ab  +  1^     a^-l^     a-b  '  V^  a-bJV^  a) 


X 

^     x  +  y      X 


a-\-h  a^^h^     a-\-  b 


- «/      \x     y) 


x  —  y     x-\- 
x-^  y     x  —  y 

a 

h      ^ 

49.  Is  the  following  a  true  statement  ^  -  =  ^  ?   What  gen- 

h 
eral  rule  aan  you  derive  from  it  ?     Apply  your  rule  to  find 

4 

the  value  of  %i- 

A 

a 

50.  Is  the  following  a  true  statement :  -  =    ?    What  gen- 
-  ^      b 


eral  rule  can  you  derive  from  it?     Apply  the  rule  to  find 
the  value  of  |. 


51.    Show  that 


a.„)(.  +  3(,+|)(..i)=(.+i..+i 


Reduce  each  of  the  following  fractions  to  a  mixed  number 
before  performing  the  indicated  additions  or  subtractions: 


52. 


x-1         x-{-l 


OS.     ■ -j • 

x-^1  x—1 


CHAPTER  VI 
FRACTIONAL   AND   LITERAL   EQUATIONS 

Fractional  Equations 

144.  Definition.  A  rational  equation  in  one  unknown 
number  which  is  not  integral  with  respect  to  that  number 
must  contain  the  unknown  number  in  the  denominator  of 
one  or  more  terms.  Such  an  equation  is  called  a  frac- 
tional equation  in  one  unknown  number. 

11  3  a:  2  5 

Thus,  -  +  -  =  2  and -\ ^^r  +  -  =  0  are  fractional  equations. 

X      a  a;  —  2a:  +  22 

145.  Clearing  an  equation  of  fractions.  From  a  frac- 
tional equation  in  which  each  fraction  has  been  reduced 
to  lowest  terms,  an  integral  equation  can  be  derived  by 
multiplying  both  members  of  the  equation  by  the  lowest 
common  denominator  of  all  its  fractions.  This  process  is 
called  clearing  the  equation  of  fractions. 

146.  Solution  of  a  fractional  equation.  The  solution  of 
a  fractional  equation  is  found  by  solving  the  integral 
equation  obtained  from  it.     Hence: 

Rule.  To  solve  a  fractional  equation,  in  which  each  frac- 
tion is  in  its  lowest  terms : 

1.  Clear  the  equation  of  fractions  hy  multiplying  both  mem- 
bers by  the  lowest  common  denominator  of  all  the  fractions. 

2.  Solve  the  resulting  integral  equation. 

3.  Test  the  roots  by  substituting  them  in  the  given  equation. 

191 


192  ELEMENTARY  ALGEBRA 


ILLUSTRATIVE   EXAMPLES 

1.    Solve  -^- — =  0. 

a; +  3       3  a;  4-1 

Solution.  2^+J:_|£jI^=:0.  (1) 

Clearing  (1)  of  fractions  by  multiplying  both  members  by 
(a:  +  3)(3x  +  l), 
(2a;  +  l)(3a:  +  l)-(x  +  3)(5a:  -  2)=  0.  (2) 

Performing  the  indicated  operations  in  (2), 

a;2_8a:  +  7  =  0.  (3) 

Factoring,  (a:  -  7)(a:  -  1)=  0.  (4) 

Equating  (x  -  7)  to  0,  a;  -  7  =  0.  (5) 

Equating  (a;  -  1)  to  0,  a:  -  1  =  0.  (6) 

Solving  (5),  a;  =  7.  (7) 

Solving  (6),  a:  =  L  (8) 

rtx.     ^  2  a:  4-  1      5  X  —  2      rt  ^^x 

Check.  ^  - =  0.  (9) 

a:  +  3       3ar  +  l  ^  ^ 


Substituting  7  for  x  in  (9), 

2x7+1      5x7-2 


0.  (10) 


7+3  3x7+1 

Simplifying,  I  "  I  =  «•  (H) 

Substituting  1  for  x  in  (9), 

2x1  +  1  5x  1-2^Q  ..^ 

1  +  3  3x1  +  1        •  ^  ^ 

Simplifying,  f  -  f  =  0-  (13) 

2      Solve  ^-^^  +  ^  16(a:+l)        _^ 

Solution.  4zl|£±1_       16{^_L11^=0.  .  (1) 

ar2  +  3  a;  -  4      5  a:^  +  46  a:  +  41  ^  ^ 

Factoring,       _(^-^D! IH^^l)        ^^  ^^^ 

^'       (a:  -  1)  (a:  +  4)      (5  a:  +  41)  (a:  +  1)  ^  ^ 

Reducing  the  fractions  to  lowest  tferms, 

a:-l  16 


a:  +  4      5  a:  +  41 


=  0.  (3) 


FRACTIONAL  AND  LITERAL  EQUATIONS        193 
Clearing  (3)  of  fractions,  and  dividing  by  5, 


x2  +  4a:-  21  =  0. 

(4) 

Factoring, 

(a:  +  7)(a'-3)=0. 

(5) 

Equating  (x  +  7)  to  0, 

X  +  7  =  0. 

(6) 

Equating  (x  -  3)  to  0, 

a:  -  3  =  0. 

(7) 

Solving  (6), 

x  =  -7. 

(8) 

Solving  (7), 

Check.                   I"'^-^! 
x^  +  3x  -4: 

x  =  S. 
16(.  +  1)        _o 
5  x^  +  46  j;  +  41 

(9) 
(10) 

Substituting  -  7  f  or  ar  in  (10), 

(_  7)2- 2(- 7)4-1                16(-7+l)            _Q 

(11) 
(12) 

(_7)2_,.3(_7)_4      5(_7)2+46(_  7)+4i 

Simr^lifvinp-               49  +  14  +  1                   -96 

Simplifying,            49  _  21  -  4 

245  -  322  +  41 

or, 

f-f-0- 

(13) 

Substituting  3  for  x  in  (10), 

(3)2  -  2  X  3  +  1 

16(3  +  1)            _  Q 

(3)2  +  3x3-4      5x 

32  +  46  X  3  +  41 

0.       1-^  •                       9-6  +  1 
Simplifpng,                 g^g_^ 

64            _Q 

45  +  138  +  41 

or 

1-1=0. 

Note.  In  certain  exceptional  cases  the  integral  equation  obtained 
by  clearing  a  given  equation  (in  which  each  fraction  is  in  its  lowest 
terms)  of  fractions  by  multiplying  both  members  by  the  lowest  com- 
mon multiple  of  the  denominators,  is  not  equivalent  to  the  given 
equation.  It  is  important,  therefore,  that  each  root  of  the  integral 
equation  should  be  checked  by  substituting  it  for  the  unknown  num- 
ber in  the  given  fractional  equation. 

The  way  in  which  these  exceptional  cases  arise  may  be  seen  from 
a  consideration  of  a  particular  example.  Thus,  let  it  be  required  to 
determine  the  value  of  x  from 


•r  +  1  1         1        _      1 

X            x(x  —  1)       X  —  1 

Clearing  of  fractions, 

(a;2-l)+  l=x. 

Simplifying, 

x^  -  X  =  0. 

Hence, 

X  =  OoTh 

194  ELEMENTARY  ALGEBRA 

On  substituting  the  value  0  for  x  in  the  given  equation,  the  first 
fraction  becomes  -,  which  is  without  meaning,  and,  therefore,  this 

value  of  X  does  not  satisfy  the  given  equation.  In  like  manner,  the 
second  root  of  the  integral  equation  does  not  satisfy  the  given 
equation. 

By  writing  the  given  equation  in  the  form 


X 

+  1+ 

X 

1 

x(x  — 

1)         X 

1 
-1 

=  0 

and  simplifying,  we 
Simplifying, 

obtain 

x^- 
x(x- 

■  X 

-1) 

1  = 

=  0. 
=  0. 

Now  this  is  an  impossible  number  relation,  and  the  reason  for  the 
non-existence  of  any  value  of  x  which  will  satisfy  the  given  equation 
is  evident.  In  fact,  although  the  given  relation  is  expressed  in  the 
form  of  an  equation,  the  expression  is  not  actually  an  equation. 

EXERCISE  72 

Solve  the  following  equations  and  check  each  solution  : 
1.   1  =  2. 

X 

*■  -  +  -  =  !• 

X       X       6 

_       1         2 


x  —  2     X 


9.5  + 


X      S  —  X 


11.    -^ ^— =  0. 

X-\-4:  1X-\-l 

15  "  5 


2.   2=3. 

3.    8  =  1 

X 

X 

'■  i- 

2x 
3 

6.    2=1 
8x     4 

8. 

2 
x  +  2 

.§  =  0. 

X 

10. 

4      , 

^     -0 

x-^  ' 

x  +  1-^- 

12. 

2 
x+S 

%=0. 
x  —  1 

14. 

5 
x-^1 

^     -0 

x-1-^- 

1A 

5 

1 

2a:-3     ^x-2  2x-]-b     2x+l 


FRACTIONAL  AND  LITERAL  EQUATIONS       195 

17.    — - — -f — - —  =  0.         18.    — ?_+— i —  =  0. 
Sx-2      5x  +  2  2x-h2     Sx-S 

19.  -^ L-  =  0. 

Sx-\-2     x-\-l 

on  ^    ^         1  4  1 

20. h 


27. 


28. 


so: 


2x     Sx-9     Zx 
21       l+i=         ^ 


3  a;  6     x{x-T) 

22.   ?+l  =  |-  23.    3^  +  2  =  ^. 

^^     3a;-4  ,  x-\  .             o«        2a:         3a:-5 

24. =  1.  25.    = 

2x^-\     a:+4  3a;-8       x-\-2 

X       ,  x-4:  2x^-1 

26. -  + 


2x-l      x+S      2a^-a;-21 

3a:-2      3a:  +  l        29  a;  +  19    ^^ 
x-\-l        2a:-l      2a:2^.a;_l 

4(a:4-2)        7(a:-l)  3(a:+l)     ^n 

2^_l         a^-a^-2      ar2-3ar-h2 


29.    ^_4._1_4.      ^  6a.»  +  30a:  +  ll 


a;+l      a:4-2      a;  +  3      (a:H- l)(a:  +  2)(a:4- 3) 
a^  +  1      a;  4- 2      a;  +  5      a;  +  6 


a:  +  2      a;+3      x -i- 6      x  +  1 
Suggestion.     Simplify  each  member  before  clearing  of  fractions. 

31.  A  can  perform  a  piece  of  work  in  two  thirds  of 
the  time  in  which  B  can  ;  B  can  perform  it  in  one  half  of 
the  time  in  which  C  can  ;  they  all  can  perform  it  in  22 
days.  Find  the  time  in  which  each  alone  can  do  the 
work. 


196  ELEMENTARY  ALGEBRA 

,  32.  A  and  B  working  together  can  do  a  certain  piece 
oi  work  in  6  days.  A  working  alone  can  do  the  same 
work  in  10  days.     In  what  time  can  B  alone  do  the  work? 

Suggestion.     Let  x  =  the  number  of  days  required. 

Then,  -  =  the  part  B  can  do  in  1  day. 

33.  What  number  must  be  added  to  each  term  of  the 
fraction  ^  so  that  the  resulting  fraction  shall  equal  §? 

34.  What  number  must  be  subtracted  from  each  term 
of  the  fraction  l|  so  that  the  resulting  fraction  shall  equal 

35.  Find  a  proper  fraction  whose  numerator  and  denomi- 
nator differ  by  1,  and  such  that  the  result  of  adding  i^  to 
the  fraction  is  double  the  result  of  subtracting  the  fraction 
from  2. 

36.  A  can  do  a  piece  of  work  in  10  days  and  B  can 
do  it  in  8  days.  In  what  time  can  they  do  it  working 
together? 

37.  A  certain  pipe  can  fill  a  cistern  in  10  hours  and 
another  can  fill  it  in  12  hours.  In  what  time  can  they  fill 
it  if  both  pipes  run  together? 

38.  A  certain  fraction  is  equal  to  |.  When  its  numer- 
ator is  diminished  by  5  and  its  denominator  by  8,  the 
resulting  fraction  is  equal  to  |.     Find  the  fraction. 

Suggestion.     Let  the  fraction  be  denoted  by  ^ — 

X 

39.  A  can  do  a  certain  piece  of  work  in  3  days,  B  can 
do  it  in  4  days,  and  C  can  do  it  in  6  days.  In  what  time 
can  they  do  it  working  together? 

40.  Three  pipes  are  connected  with  a  certain  reservoir. 
The  first  pipe  can  fill  it  in  2  hours,  the  second  in  5  hours, 


FRACTIONAL  AND  LITERAL  EQUATIONS       197 

and  the  third  can  empty  it  in  10  hours.  In  what  time 
will  the  reservoir  be  filled  if  the  three  pipes  are  set  to  flow 
at  the  same  time? 

Literal  Equations 

147.  Numerical  equation.  A  numerical  equation  is  an 
equation  in  which  all  the  known  numbers  are  expressed 
by  numerals. 

Thus,  6a:  —  l=2x4-3isa  numerical  equation. 
Remark.     The  equations  which  have  been  considered  in  the  pre- 
ceding pages  are,  in  general,  numerical  equations. 

148.  Literal  equation.  A  literal  equation  is  an  equation 
which  involves  one  or  more  known  literal  numbers. 

Thus,  ax  -f  6a;  +  c  =  0  is  a  literal  equation. 

Remark.  In  section  69  it  was  stated  that  in  algebra  known  num- 
bers are  usually  represented  by  the  first  letters  of  the  alphabet. 

149.  Solution  of  a  literal  equation.  A  literal  equation 
in  one  unknown  number  is  solved  in  the  same  manner  as 
a  numerical  equation,  but  the  roots  of  a  literal  equation 
are  algebraic  expressions  in  the  known  numbers. 

The  known  numbers  in  a  literal  equation  are  called  such  to  dis- 
tinguish them  from  the  unknown  numbers.  In  an  equation  it  is 
sometimes  convenient  to  consider  one  of  the  literal  numbers  as  the 
unknown  and  at  other  times  another. 

Thus,  from  the  formula  A  =  ab,  which  expresses  the  area  of  a 
rectangle  in  terms  of  its  base  and  altitude,  A,  a,  or  6  can  be  found 
when  the  other  two  are  known. 

ILLUSTRATIVE  EXAMPLES 

X—  b      X—  a      a^-\-l^ 


1.    Solve  the  equation 


a—  h      a-\-  b      cfi  —  J)^ 


Solution.  x^^_x_-a^a^  +  h\ 

a-ba  +  ba^-b^  ^  ^ 


198  ELEMENTARY  ALGEBRA 

Clearing  (1)  of  fractions, 

(a  +  b)(x-b)-(a-  b)(x  -  a)  =  a^  4-  b\  (2) 

Performing  the  indicated  multiplications, 

ax  +  bx  -  ab  -  b^  -  ax  +  bx  +  a^  -  ab  =  a^  -\-  b^.  (3) 

Combining  like  terms, 

2bx-^a^  -2ab  -b^  =  a^-\-  b^.  (4) 

Transposing  terms,  2bx  =  2ab  +  21^.  (5) 

Dividing  by  2  6,  x  =  a  +  b.  (6) 


2.    Solve  the  equation 


x-{-a      x-\-b      x—a-\-b. 

Solution.  -^ ^  =     ""-^    .  (1) 

X  -\-  a     X  +  b      X  —  a  -\-  b 


Combining  terms  in  the  first  member, 

(g  —b)x       _     a  —  b 
(x  +  a)(x  +  b)      X  —  a  +  b 

Dividing  both  members  of  (2)  by  (a  —  6), 

X  1 


Check  :  Substituting for  x  in  (1), 


a-b 


!+«  -|+*  -!-«+* 


(2) 
(3) 


(x  -{■  a)(x  -{■  b)      X  —  a  -{■  b 
Clearing  (3)  of  fractions, 

x^  —  ax  -\-  bx  =  x^  -{-  ax  +  bx  +  ab.  (4) 

Transposing  in  (4)  and  uniting  like  terms, 

2ax  =  -  ab.  (5) 

Dividing  both  members  of  (5)  by  2  a. 


-  =  -|.  (6) 


(7) 


Simplifying,  f^Ta^  f^a  <«> 


FRACTIONAL  AND  LITERAL  EQUATIONS       199 

3.    What  number  must  be  added  to  both  numerator  and 
denominator   of   the    fraction    —    so   that   the    resulting 

fraction  shall  be  equal  to  -  ? 

P 

Solution.     Let  x  denote  the  required  number. 
From  th6  conditions  of  the  problem, 

I    +    X     _n  .y. 

(2) 
(3) 

(4) 


m-\-  X     p 

Clearing  (1)  of  fractions, 

Ip  -\-  px  =  nm  +  nx. 

Transposing  in 

(2)  and  uniting  like  terms, 
(p  —  n)x  =  nm  —  Ip. 

Dividing  both  members  of  (3)  by  {p  -  n), 

^_nm-lp 

p-n 

Check. 

1  +  X                 p-n 

wi  +  ^      ^   ^  nm-  Ip 

p-n 

_  pi  —  In  +  nm  - 

'¥ 

pm  —  nm  +  nm 

-Ip 

_  n(m  —  I)  _  n 

p{m  -  1)      p 

Application  of  formula.     Substitute  Z  =  5,  w  =  6,  n  =  16,  p  =  17, 
in  formula  (4), 

nm  —  Ip 


p- 

-  n 

6  X 

16- 

5x17 

17- 

16 

96- 

-85, 

or  11. 

200  ELEMENTARY  ALGEBRA 

EXERCISE  73 

'  Solve  the  following  and  check  all  roots: 


1. 

X  —  h  =  a. 

2. 

^  =  b. 

3. 

1_1 
X     a 

4. 

a 
ax=  b. 

5. 

2x-\-ax  =  l. 

6. 

ax  —  x-^bx=sc. 

."■■'  7. 

ax-}-  bx  —  x  =  0. 

8. 

X 

9. 

X      a 

10. 

1 

11. 

ax-{-bx  =  1. 

12. 

ax-l  =  2-bx 

13. 

1         a 
x-1      2 

• 

14. 

ax—bx-^(c-^  d)x  =  a 

-b-^c-\-d. 

15. 

ax  —  b  =  d  —  ex. 

16. 

^-5  =  ^ 

a-{-b  _     1  aa:      bx      ex      ^ 

19.  2  aa:  —  3  6a;  =  2  5a;  —  3  aa;  -h  <?. 

20.  (a  +  ^>-l>-(a-6+l)  =  (a-h6-l)-(a-6+2)a;. 

21.  a(a:  -  a)  +  6(a;  -  6)  =  3  «a;  -  (a  +  5)2. 

22.  a(^x  +  5)  +  5(a:  +  a)  =  2  aa:  +  52  -  (a  -  5)2. 

23.  (a:  +  a)2  =  4a2  4-(a;-a)2. 

24.  (a;-a)(a;-5)-(a;-(?)(a;-(^)=0. 

25.  a  +  -=e. 

26.  -  +  -  =  <? 

XX  X 


FRACTIONAL  AND  LITERAL  EQUATIONS       201 

27.  «_*  +  £  =  !. 

XXX 

28.  fL±£+*±£+fL±*  =  o. 

ax  hx  ah 

29.  £^^  +  *±f  +  l  +  U0. 

c^x         y^x        a      h 

^^     2      h      a-2x 

30.  — —  = 

b     X     2h^bx 

31.  £zif  +  £+*  =  2. 
x  +  a      x^o 

32. 1 =  a^  4-  6. 

x  —  aax—o 

-,-,  ^        .        ^  .1 

33. + —  =  a  H-  6. 

1  —  aa;      1  —  6a; 

34.  ^+        *   ■ 


x-\-  b~  2x-\-b-\-e 


ac  —  x      ah  —  X      ah—  x 
35. 

36.    ah:(hx-l)+h^x(ax-l)  =  (a-^b')(ax-t)(ibx-'l}, 

EXERCISE  74 

1.  A=  ah. 

(a)  Solve  for  a. 
(6)  Solve  for  6. 

2.  (7=2  7rr.     Solve  for  r. 

3.  The  volume  of  a  prism  is  expressed  by  the  formula 

(a)  Solve  for  B, 
(5)  Solve  for  ^. 


202  ELEMENTARY  ALGEBRA 

4.  The  volume  of  a  rectangular  parallelepiped  is  ex- 
pressed by  the  formula  V=  ahc. 

(a)  Solve  for  a. 
(6)  Solve  for  5. 

(c)  Solve  for  e. 

(d)  Find  a  when  F=  100,  b  =  10,  and  c  =  6. 

5.  The  lateral  area  of  a  regular  pyramid  is  expressed 
by  the  formula  S  =  ^  L  x  P. 

(a)  Solve  for  L, 
(h)  Solve  for  P. 
(c)  Find  L  when  i^  =  40  and  P  =  8. 

6.  The    volume    of   a   pyramid   is   expressed   by   the 
formula  V=\B  x  H, 

(a)  Solve  for  B, 
(h)  Solve  for  ^. 
(c)  Find  ^  when  r=  63  and  j5  =  15. 

7.  The  lateral  area  of  a  cylinder  of  revolution  is  ex- 
pressed by  the  formula  S  =  2  m-RH. 

(«)  Solve  for  R, 
(5)  Solve  for  ^. 
(c)  Find  R  when  aS'  =  24  and  ir=  4. 

8.  The  lateral  area  of  a  cone  of  revolution  is  expressed 
by  the  formula  S  =  tt  RL. 

(a)  Solve  for  R, 
.    (5)  Solve  for  X. 

(c)  Find  aS'  when  i?  =  4  and  i  =  5. 

(d)  Find  L  when  aS'  =  15  and  i2  =  3. 

9.  A  =  \(h  H-  B^a  is  the  formula  stating  the  area  of  a 
trapezoid. 

(a)  Solve  for  a. 

(h)  Solve  for  (h  +  J?). 


FRACTIONAL  AND  LITERAL  EQUATIONS       203 

(6?)  Solve  for  h. 

(cT)  Find  a  froAi   the   following   data  :    ^  =  95, 
^  =  20,  5  =  18. 

10.    V  =  ^*  is  a  formula  from  physics, 
(a)  Solve  for  t. 
lb)  Find  t,  when  g  =  32.16  and  v  =  80.4. 


11.  h  = z—  is  a  formula  from  engineering. 

1.2b  r 

(a)  Solve  for  w. 
(h)  Solve  for  r. 

12.  -  =  -  +  -  is  a  formula  from  physics. 
fab 

(a)  Solve  for/. 

(6)    Solve  for  a. 

Prt 

13.  ^  =  P  +  — —  is  a  formula  from  arithmetic. 

(a)  What  do  ^,  P,  r,  and  ^,  respectively,  repre- 
sent in  the  preceding  formula  ? 
(6)  Solve  for  P. 
(c)  Solve  for  t. 

id)  Solve  for  ^. 

14.  /=—  is  a  formula  used  in  electrical  work. 

R 

(a)  Solve  for  ^. 
(5)  Solve  for  E. 

(c)  Find  7  when  i:  =  200  and  i2  =  250. 

(d)  Find  T;  when  7=  1.5  and  i2  =  200. 

(e)  Find  R  when  7=  .4  and  ^  =  120. 

15.  The  formula  for   converting   a  temperature  of  F 
degrees    Fahrenheit    into   its   equivalent    of    0   degrees 


204  ELEMENTARY  ALGEBRA 

Centigrade  is  C  =  -  (F-  32).     Express  F  in  terms  of  C, 

and  compute  F  when  : 
(a)  (7=20. 
(6)   (7=30. 
(0   (^=27. 

16.  What  must  be  added  to  a;  +  a  to  make  y  —  h? 

17.  What  is  the  cost  of  3  oranges  if  a  oranges  cost  e 
cents  ? 

18.  Divide  the  number  n  into  four  parts  such  that,  if  a 
is  added  to  the  first,  a  subtracted  from  the  second,  the 
third  multiplied  by  a,  and  the  fourth  divided  by  a,  the  re- 
sults will  be  equal.  What  are  the  results  if  7i  =  10,  and 
a  =  l? 

19.  If  my  age  is  such  that  in  n  years  I  shall  be  a  times 
as  old  as  I  was  m  years  ago,  what  is  my  age  ? 

20.  If  each  side  of  a  square  were  n  feet  longer,  its  area 
would  be  p^  square  feet  greater.  Find  the  length  of  its 
side. 

21.  How  many  pounds  of  coffee  worth  a  cents  a  pound 
must  be  mixed  with  b  pounds  worth  e  cents  a  pound  so 
that  the  mixture  may  be  worth  d  cents  a  pound  ? 

22.  Find  the  number  the  sum  of  whose  ath  and  bth 
parts  is  c.  What  is  the  number  when  a  =  5,  6  =  7,  and 
c=12? 

W 


23.    t  = 


•\  V  J    V 


W-\-  w' 

(a)  Solve  for  W, 

(b)  Solve  for  w. 
((?)   Solve  for  V. 

(c?)  Calculate  Tfwhen  t  =  .0019,  w  =  2,  V=  -4, 
V  =  100,  a  =  0.1. 


Blaise  Pascal  v  1 623- 1 662)  was  born  at  Clermont  and  died  at 
Paris.  Perhaps  no  man  ever  displayed  greater  natural  genius 
than  Pascal.  His  contributions  to  mathematics  were  not  extensive, 
but  his  name  will  always  be  mentioned  in  connection  with  the 
arithmetical  triangle.  Much  of  his  mathematical  work  was  done 
when  he  was  a  boy.  His  essay  on  conic  sections  was  written  in 
1639,  when  he  was  but  sixteen  years  old. 


CHAPTER  VII 
SYSTEMS   OF   LINEAR   EQUATIONS 

150.  An  equation  may  involve  more  than  one  unknown 
number. 

Thus,  the  equation  2x  +  3  y  =  5  contains  two  unknown  numbers, 
and  the  equation  ax  -\-  by  +  cz  +  d  =  0  contains  three. 

151.  Degree  of  an  equation.  By  the  degree  of  a  rational 
integral  equation  in  two  or  more  unknowns  is  meant  its 
degree  with  respect  to  the  unknown  numbers. 

Thus,  2  X  +  3  y  =  5  is  an  equation  of  the  first  degree  in  the  two 
unknown  numbers  x  and  y ;  3  x^  —  2  xy  =  1  is  an  equation  of  the 
second  degree ;  2  x^y  +  3  s;^  =  5  is  an  equation  of  the  third  degree. 

152.  Linear  equation.  An  integral  equation  of  the  first 
degree  in  any  number  of  unknowns  is  called  a  linear  equa- 
tion, or  a  simple  equation. 

Thus,  2  a:  =  6  and  2  x  -\-  3  y  =  12  are  linear  equations. 

153.  Solution  of  an  equation.     A  solution  of  an  equation 

in  more  than  one  unknown  is  afi^  set  of  values  of  the  un- 
known numbers  which  satisfies  the  equation. 

Thus,  the  linear  equation  2  x  -\-  3  y  =  5  has  among  other  solutions 
the  following  sets : 

x  =  l,  y=l;    x  =  0,  y  =  ^;    x  =  ^,  y  =  0. 

154.  Number  of  solutions  of  an  equation.  The  number 
of  solutions  of  an  equation  in  more  than  one  unknown  is 
not  determinate. 

205 


206  ELEMENTARY  ALGEBRA 

Thus,  in  the  equation  2  r  +  3  ^  =  5,  we  may  assign  any  value  to  x 
and  solve  the  resulting  linear  equation  for  y ;  for  example : 

Let  .  a:  =  7 ; 

then,  14  +  3  3/  =  5, 

whence,  y  =  —  3. 

A  solution  of  the  equation  2  a:  +  3  y  =  5  is,  therefore,  a:  =  7,  y  =  —  3. 
Remark.     It  is  because  the  number  of  solutions  of  an  equation  in 
more  than  one  unknown  is  not  determinate  that  such  an  equation  is 
called  an  indeterminate  equation. 

ILLUSTRATIVE   EXAMPLES 

Find    four    solutions    of    the    indeterminate    equation 
2a;-|-3y=10,  and  check  each  solution  found. 

Solution.  2  X  4-  3  3/  =  10.  (1) 

Solving  (1)  for  y  in  terms  of  x, 

y  =  '^^-  (2) 

Assigning  any  four  values  to  x,  as  0,  1,  -  1,  2,  we  have, 
10  -  2  X  0     10 


when  a;  =  0, 
when  a;  =  1, 


3  3  ' 

10  -  2  X  1      8 


3  3' 

whenar  =  -l,  ^  ^  10  -  2(- 1)  ^  ^. 

3 

when  a:  =  2,  V  =  ^^  ~  ^  ^  ^  =  2. 

'  ^  3 

Four  solutions  of  2  x  +  3  y  =  10  are,  therefore,  (0,  ^2-),  (1,  |), 
(  —  1,  4),  (2,  2),  in  each  of  which  the  value  of  x  stands  first. 
Check.  2  a;  +  3  y  =  10. 

Substituting  0  for  x  and  ^^-  for  y  in  the  given  equation, 
2x  0  +  3(J^)=10. 

Substituting  1  for  x  and  J  for  y, 

2  X  1  +  3  X  I  =  10. 


SYSTEMS  OF  LINEAR  EQUATIONS  207 

Substituting  —  1  for  x  and  4  for  y, 

2(-l)+3  x4  =  10. 
Substituting  2  for  x  and  2  for  y, 

2x2  +  3x2  =  10. 

EXERCISE  76 

Find   four   solutions    of    each    of    the    following    inde- 
terminate equations  and  check  each  solution  found. 
1.    y  —  2x.  •  2.    x-\-y  —  4:. 

Z.    x-\-1y  =  l.  4.    2x  —  Sy  =  5, 

5.    3  2:  +  3«/  =  5.  €.    y  =  x  +  l. 

7.    y  =  mx.  B.    y  =  mx  +  h. 

155.  Independent  equations.  Two  linear  equations  in 
two  or  more  unknowns  are  called  independent  equations  if 
each  has  solutions  which  are  not  solutions  of  the  other. 

Thus,  X  -\-  y  =  ^  and  j-  —  y  =  2  are  two  independent  linear  equa- 
tions, as  a-  =  8  and  y  =  0  is  a  solution  of  the  first  equation  which  is 
not  a  solution  of  the  second ;  and  x  =  2  and  y  =  0  is  a  solution  of 
the  second  equation  which  is  not  a  solution  of  the  first. 

156.  Dependent  equations.  Two  linear  equations  in  two 
or  more  unknowns  are  called  dependent  equations  when 
every  solution  of  the  one  is  also  a  solution  of  the  other. 

Thus,  X  -{-  y  =  \  and  2  ar  +  2  y  =  2  are  two  dependent  linear 
equations. 

157.  Inconsistent  equations.  Two  equations  which  have 
no  common  solution  are  called  inconsistent  equations. 

Thus,  X  -\-  y  =  2  and  ar  +  y  =  1  are  evidently  inconsistent,  since  the 
sum  of  two  numbers  cannot  be  both  2  and  1  at  the  same  time. 

158.  Simultaneous  equations.  Two  or  more  equations 
in  more  than  one  unknown  number,  when  considered 
together,  are  said  to  be  simultaneous  if  they  have  at  least 
one  solution  in  common. 


208  ELEMENTARY  ALGEBRA 

159.  System  of  equations.  Equations  considered  to- 
gether are  called  a  system  of  equations. 

160.  Number  of  solutions  of  two  independent  linear 
equations  in  two  unknown  numbers.  Two  independent 
and  consistent  linear  equations  in  two  unknown  numbers 
are  satisfied  by  one  and  only  one  set  of  values  of  the 
unknown  numbers.  Such  a  system  of  linear  equations 
has,  therefore,  only  one  solution.     - 

Note.  The  student  in  his  later  algebraic  work  will  find  that  the 
fact  stated  in  section  160  covers  a  particular  case  which  is  included  in 
a  more  general  statement.  He  can,  however,  satisfy  himself  that 
two  independent  linear  equations  in  two  unknowns  have  no  more 
than  one  solution  in  common.  For  instance,  to  show  that  x  -\-  y  =  S 
and  X  —  y  =  4:  have  one  and  only  one  solution,  we  may  write 

x  +  y  =  S,  (1) 

x-y  =  i;  (2) 

then  by  adding  (1)  and  (2)  we  have  2  a:  =  12.  Therefore,  any  value  of 
X  which  satisfies  both  equations  must  be  such  that  when  multiplied 
by  2  the  result  is  12,  and  there  is  but  one  such  number.  Moreover,  by 
subtracting  (2)  from  (1),  we  have  2  y  =  i,  and  there  is  but  one  value  of 
y.  It  will  be  made  evident  in  the  course  of  this  chapter  that  a  similar 
proof  based  on  the  theory  of  equivalent  systems  of  equations  holds 
for  the  general  equations  of  the  form  ax  +  by  =  c  and  mx  +  ny  =  p. 

Remark.  Two  systems  of  equations  in  two  or  more  unknown 
numbers  are  said  to  be  equivalent  when  every  solution  of  the  first 
system  is  a  solution  of  the  second  and  every  solution  of  the  second 
is  a  solution  of  the  first. 

161.  Method  of  solution.  Two  independent  linear  equa- 
tions in  two  unknowns  are  solved  by  combining  these 
equations  in  such  a  way  that  there  results  a  simple  equa- 
tion in  one  unknown.  One  of  the  unknowns  does  not 
appear  in  the  resulting  equation ;  it  has  been  eliminated. 
The  process  by  which  the  unknown  is  eliminated  is  called 
elimination. 


SYSTEMS  OF  LINEAR  EQUATIONS  209 

162.  Elimination  by  addition  or  subtraction.  The  solu- 
tion of  a  system  of  two  linear  equations  by  employing  the 
method  of  elimination  by  addition  or  subtraction  is  ex- 
plained in  the  following : 

ILLUSTRATIVE   EXAMPLES 

1.    Solve  the  system  |!^  +  ^^  =  f'  « 

Solution.  Multiplying  (1)  by  2  and  (2)  by  3  so  that  the  co- 
efficients of  y  in  the  two  equations  shall  have  the  same  absolute 
value, 

4  a:  +  6  3/  =  10.  •  (3) 

9x-6y  =  3.  (4) 

Adding  (3)  and  (4),  13  a:  =  13.  (5) 

Dividing  by  13,  a:  =  1.  (6) 

Substituting  the  value  of  x  in  (1), 

2x1  +  33/  =  5.  (7) 

Transposing  and  uniting  terms, 

33^  =  3.  (8) 

Dividing  by  3,  2/  =  1.  (9) 

Therefore,  the  solution  of  equations  (1)  and  (2)  is  a:  =  1,  y  =  1. 
Check.     Substituting  1  for  x  and  1  for  y  in  equations  (1)  and  (2), 
we  have, 

2x1  +  3x1  =  5, 
3x1-2x1  =  1. 

Remark.  In  the  solution  of  example  1,  equations  (1)  and  (2) 
were  replaced  by  the  equivalent  system  (3)  and  (4)  ;  that  is,  the  solu- 
tion of  (1)  and  (2)  is  the  same  as  the  solution  of  (3)  and  (4).  The 
actual  operations  of  addition,  subtraction,  multiplication,  and  division, 
here  as  elsewhere  in  solving  equations,  depend  on  the  principles'stated 
in  section  25. 


Solve  the  system 

3a7+-2y  _  Zx-ly  ^  ^  _  2^-21  x 


12 


(1) 


bx-Zy     52;+3y^^      x-\-y  .^^ 

2  3  6     '  ^  ^ 


210  ELEMENTARY  ALGEBRA 

Solution.  Clearing  equations  (1)  and  (2)  of  fractions,  trans- 
posing, and  uniting  terms,  we  have  the  equivalent  system, 

81  ar  -  74  3^  =  -  60.  (3) 

Qx-  y  =  9.  (4) 

Multiplying  both  members  of  (4)  by  74,  so  that  the  coefficient  of  p 

in  the  resulting  equation  shall  be  equal  to  the  coefficient  of  y  in  (3), 

444  ar  -  74  y  =  666.  (5) 

Subtracting  each  member  of  (3)  from  the  corresponding  member 

of  (5), 

363  X  =  726.  (6) 

Dividing  by  363,  x  =  2.  (7) 

Substituting  the  value  of  x  in  (4), 

12  -  2/  =  9.  (8) 

Solving  (8)  for  y,  y  =  S.  (9) 

Therefore,  the  solution  of  equations  (1)  and  (2)  is  x  =  2,  y  =  3. 

Check.  Substituting  2  for  x  and  3  for  y  in  equations  (1)  and  (2), 
we  have 

3x2  +  2x3     3x2-2x3_i      2  x  3  -  21  x  2         ^^m 
•        3 5  "  12  '       A^"^ 

5x2-3x3      5x2  +  3x3  ^  g      2  +  3  ,, , . 

2  3  6     '  ^    ^ 

Simplifying  (10),  4  =  4.  (12) 

Simplifying  (11),  ^^  =  ^K  (13) 

^  ,        1  ,         \ax-\-by=:m^  (1) 

3.    Solve  the  system  {  ,  ;^. 

•^  \bx  —  ay  —  n.  (2) 

Solution.  Multiplying  both  members  of  (1)  by  a  and  both  mem- 
bers of  (2)  by  b  in  order  that  in  the  two  resulting  equations  the  coeffi- 
cients of  y  may  have  the  same  absolute  value, 

a^x  +  ahy  =  am.  (3) 

6%  —  ahy  =  hn.  (4) 

Adding  (3)  and  (4),  (a^  +  h^)x  =  am -{■  hn,  (5) 

Dividing  by  a2  + 62,  :,  =  ^_±^.  (6) 


SYSTEMS  OF  LINEAR  EQUATIONS  211 

Substituting  in  (1)  the  value  of  x  as  found  in  (6), 

ahn  +  abn  ,    ,  .  ^r,^ 

~^^-^  +  by  =  m.  (7) 

Clearing  (7)  of  fractions, 

a^in  -t-  abn  +  (a^  +  b^)bt/  =  wm^  -f  inb^.  (8) 

Transposing  and  collecting, 

(a2  +  b^)bt/  =  mb^  -  abn.  (9) 

Dividing  both  members  of  (9)  by  6, 

(o2  +  b^)y  =  mb-  an.  (10) 

Dividing  (10)  by  a^  +  b\ 

mb  —  an  x-  - . 

Therefore,  the  solution  of  equations  (1)  and  (2)  is 

nm  +  bn            mb  —  an 
x= ■ ,  u  = • 

Remark.  The  value  of  i/  might  also  have  been  found  as  the  value 
of  X  was  found ;  namely,  by  making  the  coefficients  of  -r  alike,  com- 
bining the  resulting  equations  so  as  to  eliminate  x,  and  solving  the 
resulting  equation  for  i/. 

From  the  solutions  of  illustrative  examples  1,  2,  and  3 
we  may  infer  the  following : 

Rule.  To  eliminate  an  unknown  number,  as  «/,  from  two 
simultaneous  linear  equations  hy  addition  or  subtraction,  mid- 
tiply,  if  necessary,  the  members  of  one  or  both  equations  by 
such  a  rmmber  or  such  numbers  as  will  make  the  absolute 
values  of  the  coefficients  of  y  in  the  two  equations  the  same  ; 
then  add  or  subtract  the  corresponding  members  of  the  two 
resulting  equations  according  as  the  coefficients  of  y  in  the 
equations  have  opposite  signs  or  the  same  sign. 


212 


ELEMENTARY  ALGEBRA 


1. 


3. 


7. 


9. 


11. 


13. 


15. 


17. 


19. 


21-     1  -  13  a;  +  11  «/  =  2. 


23. 


Sx  ,  2y      . 

X        1 

^       2 


2. 


EXERCISE 

Solve  the  following  systems 
the  solution  of  each  : 

x-\- 1/  =  6, 
2^=3. 

f2:r+3y  =  7, 
\3x-S^=:U, 

|3^  +  2y  =  0. 
ll2x  +  4i/  =  S, 

Sx-^S^  =  1. 

17«/  +  22  =  19, 

15^-7;2=8. 

3a;  +  5?/  =  9, 

2a;  +  5?/=5. 

4  a; -3?/ =  16, 

14a;4-5^  =  25. 

4m4-3jt?  =  17, 

5m  —  4:p  =  2, 

2  w  -  3  V  =  4, 

5v-3i^=-9. 

ll:r  +  12y  =  l, 

17rr-l-19y  =  2. 


76 


of  equations,   and  check 


8. 


10. 


12. 


14. 


16. 


18. 


20. 


22. 


24. 


a;  +  ^  =  10, 
x  —  ^  =  4. 
5a:4-73/=17, 
5a;+ 3^  =  13. 
6a;4-3i/  =  4, 
82^-9^=1. 
3  a:  -  5  i/  =  9; 
y—  4^:  =  5. 
lla;-12  2/  =  13, 
6y-13a;  =  l. 
17ic  +  15?/  =  l, 
8  a; +  6^  =  1. 
3m +  2^1=  7, 
4m  —  57i=  6. 
5?^;-3^  =  2, 
7f-15i^  =  2. 
6w  +  5z^=-8, 
4w+  25«^  =  — 1. 
13a;4-17i/  =  4, 
31  a;-f  37^  =  6. 

2       3       2' 

if  +  ^  =  2. 
2       6 

5^ 
3 


3^^19 
5       30' 


256^^19 
7        5      35 


SYSTEMS  OF  LINEAR  EQUATIONS 


213 


25. 


27. 


29. 


31. 


33. 


35. 


37. 


39. 


40. 


41. 


3m^5n^2-0, 

26. 

^"        3=2' 

6m  +  57i-l  =  0. 

6i^+5^;  +  l  =  0. 

I  +  I+B.O, 

28. 

^-^  =  3, 
14     15       ' 

^-1  =  1. 
10      9 

Zx     by^^ 

2       17 
9  a:      Yly      o 

2         5 

30. 

Sx     2y_. 
17      13"    ' 

5:.-6^/  =  7. 

X  —  y  =  n. 

32. 

y  =  mx-\-e, 
y  =  lx-hd. 

x-\-ay=b. 

34        < 

-  +  !=!' 

ax-\-  y  =  c. 

«3V*           1 

6             a 

-  +  !  =  !' 
ax—hy  =  c. 

36. 

ax  -y  =  h 

[Sx-\-y=e. 

^+f=i,      • 

a      0 
x-y=\. 

38.      i 

y-l  =  m(x-l), 
m^y  -h  ma;  +  w^  H-  1  =  0 

x-\-2a      y-'2a_o 
a+1         a—1 

a, 

x-y-4:a_  x-\-y 

2a             a^-{-l 

H-y  .^— y_lQ  _a:+i/  +  a      x  —  y  —  h  ^ 
a  6  3  a  5 


|a:  +  2/=a  +  6, 
\ax—hy  =  a^—  ft^. 


42. 


I  aa;  +  by-\-c  =  0, 
joa:  +  ^-y  +  r  =  0. 


214  ELEMENTARY  ALGEBRA 

\  ah;  4-  h^i/  =  5  be. 


44. 


45. 


46. 


lb         a         \a      bj 

1  (a:  +  «/)(a2  +  ^2)=  ab(x  +  «/)  -h  2(a8  +  63). 

j  aa;  —  5^  =  ^2  _  2  a6  —  62^ 

\ax  -h  bi/  =  a^  -\-  If^. 
ax+bt/  =  a-^  b, 
(a  +  5)a^  —  (a-b)t/=2b. 


^^      Ua-b)xA-(a+b 
\ax ^  by  =  a^  —  1)^. 


ax  by    _  a^  —  a% -{- ab -{- b^ 


48.     la-^-b     a-b  a^-U^ 

I  («  ^.  h)x  -  (^2  +  52)^  =  b(a  -  b). 

163.  Elimination  by  substitution.  The  solution  of  a 
system  of  two  linear  equations  by  employing  the  method 
of  elimination  by  substitution  is  explained  in  the  following : 


Solve  the  system 


ILLUSTRATIVE    EXAMPLE 

3a:+5^  =  21,  (1) 

2  a:  -h  3  ^  =  13.  (2) 
Solution.     Solving  equation  (1)  for  the  value  of  x  in  terms  of  y, 

.  =  2JL^.  (8) 

Substituting  in  (2)  the  value  of  x  as  found  in  (3), 

42_-10^^3^^^3  ^^^ 

Solving  (4),                                 3,  =  3.  (5) 

Substituting  in  (3)  the  value  of  y  as  found  in  (5), 

21-15     o  /«^ 

x= — =  2.  (6) 

Therefore,  the  solution  of  (1)  and  (2)  is  x  =  2,  y  =  3, 


SYSTEMS  OF  LINEAR  EQUATIONS  215 

From  the  solution  of  the  foregoing  illustrative  example 
we  may  infer  the  following: 

Rule.  To  eliminate  an  unknown  number^  as  x,  from  two 
simultaneous  linear  equations  hy  substitution,  solve  one  of  the 
equations  for  x  and  substitute  the  resulting  value  of  x  in  the 
other  equation. 

EXERCISE  77 

Solve  the  following  systems  of  equations,  using  the 
method  of  elimination  by  substitution: 


5a:  +  3?/  =  ll,  ^     f3a:-22/  =  4, 


y==2x.  '    \Qx  +  by  =  ll. 

3x-y==S, 


ix—y=l, 

5x+7y  =  41.  ^'    \Sx-\-2y  =  10. 
l2y=Sx,  jx  +  y  =  5, 

\5x-^y=2.  *•    \5x-ly=l 


ax  =  by, 

(jOL  H-  b)x  -I-  (a  —  b)y  =  a-\-b, 

■^"-   ^(a-6H-2>  +  3?/  =  4a  +  26  +  2. 


^1 


ax+by=  c. 
^     ^ax+by  =  d, 
mx  -\-ny=c. 

(For  further  practice  in  solving  a  system  of  two  linear  equations 
by  employing  the  method  of  elimination  by  substitution,  one  or  more 
of  the  examples  in  exercise  76  may  be  taken.) 

164.  Elimination  by  comparison.  The  solution  of  a 
system  of  two  linear  equations  by  employing  the  method 
of  elimination  by  comparison  is  explained  in  the  following : 


216  ELEMENTARY  ALGEBRA 

ILLUSTRATIVE  EXAMPLE 

Solve  the  system         I  ^^       ^ ~    '  ^^^ 

^  (10a;H-92/  =  10.  (2) 

Solution.     Solving  (1)  for  x, 

(3) 


4-3w 


Solving  (2)  for  x,  x  =  ^^~^^'  (4) 

Equating  the  two  values  of  x  as  found  in  (3)  and  (4), 

4-3.y^lO-9y  .g. 

5  10      ■  >  ^ 

Simplifying  (5),  3y  =  2.  (6) 

Solving   (6),  2/  =  |.  (7) 

Substituting  in  (3)  the  value  of  y  as  found  in  (6), 

x  =  f  (8) 

Therefore,  the  solution  of  (1)  and  (2)  is  x  =  ^,  y  =  ^. 

From  the  solution  of  the  foregoing  illustrative  example 
we  may  infer  the  following : 

Rule.  To  eliminate  hy  comparison  an  unknown  number,  as 
x^from  two  simultaneous  linear  equations,  solve  each  of  the 
two  equations  for  x  and  equate  the  two  resulting  values. 

EXERCISE  78 

Solve  the  following  systems  of  equations,  using  the 
method  of  elimination  by  comparison  : 


2y-\-'2>x  =  b,  [3a;  +  4y=-l. 


11  x-l^y  =  1,  j2x-Sy  =  l, 


Ux-\-14:y  =  27.  I5a;  +  7y  =  46. 

5x-^2y=-l,  \llx-2y  =  l, 


Sx-\-5y  =  ll.  \Ux-\-l  y=:-65. 


■I 


SYSTEMS  OF  LINEAR  EQUATIONS  217 

1892:4-3^  =  3,  jllx-2S^  =  2, 

^'   \lSx-5y  =  -b.  123  2:  +  71^  =  236. 

(For  further  practice  in  solving  a  system  of  two  linear  equations 
by  employing  the  method  of  elimination  by  comparison,  one  or  more 
of  the  examples  in  exercise  76  may  be  taken.) 

Remark.  The  three  methods  of  elimination  considered  in  sections 
162,  163,  and  164  are  manifestly  applicable  to  any  system  of  two 
simultaneous  linear  equations.  Of  these  methods,  that  of  elimination 
by  addition  or  subtraction  is  most  frequently  employed.  However, 
when  one  of  the  equations  gives  the  value  of  one  of  the  unknowns, 
as  X,  in  terms  of  the  other,  elimination  by  substitution  may  be  used 
to  advantage. 

165.  Elimination  by  use  of  an  undetermined  multiplier. 
The  solution  of  a  system  of  two  linear  equations  by  use  of 
an  undetermined  multiplier  is  explained  in  the  following  : 


ILLUSTRATIVE  EXAMPLE 

Solve  the  system        1  !  ^  "^  ?  ^  ""  !'^'  ^1} 

^  [3a:-5i/  =  l.  (2) 

Solution.     Multiplying  (2)  by  m,  3  mx  —  5 my  =  m.  (3) 

Adding  (1)  and  (3),  (7  +  3w)a:  +(3  -  5m)y  =  17  +  m.  (4) 

Equating  the  coefficient  of  y  in  (4)  to  0,        3  —  5  m  =  0.  (5) 

Solving  (5),  ^  =  I-  (6) 

Substituting  the  value  of  m  in  (4),    (7  +  ■|)a:  +  0  •  y  =  17  +  |^.  (7) 
Solving  (7),  x  =  2.  (8) 

Substituting  the  value  of  x  in  (1)  and  solving,         y  =  I- 

Remark.  The  number  m  in  equation  (4)  of  the  foregoing  solution 
is  undetermined ;  that  is,  it  may  have  any  numerical  value  assigned 
to  it.  We  assign  such  a  value  to  m  that  the  coefficient  of  one  of  the 
unknown  numbers  shall  vanish.  It  is  evident  that  instead  of  first 
eliminating  y,  the  coefficient  of  x  might  have  been  placed  equal  to 
zero  and  the  value  of  y  determined. 


218  ELEMENTARY  ALGEBRA 

EXERCISE  79 

Solve  the  following  systems  of  equations,  eliminating 
one  of  the  unknowns  by  use  of  an  undetermined  multiplier: 


6m  +  5jt?  =  —  2. 


|4^-5y  =  22,  f 

l32:+.23/  =  5.  ^'    \ 

j7w-llv  =  26,  |a:-h2^  =  2, 

I15w-f  5t;  =  -30.  *•    1 


2a:-3y  =  54. 

\(m  -f  n)  -h  J(^  -  w)  =  2, 
I (m  4-  w)  +  f  (w  -  7i)  =  17. 

15^  +  19y  =  18, 
^'   h9a;  +  15^  =  50. 


1*^ 

IK 


■■! 


10. 


18a:+23?^==13, 
23  a: +  18  ^  =  28. 

2.5  a: +  3.7  3/ =  7.69, 

3.6  a; -2.9^  =  1.20. 

f. 05  a: +.03?^  =.011, 
1 .72  a: +  .93  2^  =  .258. 
f  aa;  +  6^  =  a^  ^  52^ 
\  a^  —  52^  =  a^  _  58^ 


166.  Special  systems  of  simultaneous  equations.  Cer- 
tain systems  of  simultaneous  fractional  equations,  which 
are  of  frequent  occurrence,  should  be  solved  by  the 
methods  already  employed  in  this  chapter.  In  such 
systems,  the  equations  are  not  cleared  of  fractions, 

ILLUSTRATIVE  EXAMPLES 


1.    Solve  the  system 


?+^=i,  (1) 

?+^  =  2.  (2) 

X     y 


SYSTEMS  OF  LINEAR  EQUATIONS  219 

Solution.     Writing  (1)  and  (2)  in  another  form, 


Equations  (3)  and  (4)  are  in  the  form  of  two  simultaneous  linear 

equations   in   the  unknown  numbers  -  and  -.     Multiplying   both 

X  y 

members  of  (3)  by  5  and  those  of  (4)  by  3, 

Subtracting  the  members  of  (6)  from  the  corresponding  members 
of  (5), 

-  =  -1.  (7) 

X 

Solving  (7)  for  a:,  a:  =  -  1.  (8) 

Substituting  in  (1)  the  value  of  x  as  found  in  (8), 

-2+?  =  l.  (9) 

y 

Solving  (9)  for  y,  y  =  l.  (10) 

Therefore,  the  solution  of  (1)  and  (2)  is  ar  =  —  1,  y  =  1. 
Check.     Substituting  —  1  for  x  and  1  for  y  in  (1)  and  (2),  we 
have,  respectively, 

-2  +  3  =  1,  ai) 

-8  +  5  =  2.  (12) 

Note.     The  system  (1)  and   (2)    whose  solution   has  just  been 
given   is   not   equivalent   to   the   system   obtained  by  clearing  (1) 

and  (2)  of  fractions ;  namely  the  system   j       ^         x  —  xy   ]^  ^     rjy^^^ 

[3y  +  ox  =  2xy} 
new  system  is  not  composed  of  linear  equations  and  it  has  other 
solutions  than  the  one   obtained  from  (1)  and  (2).     For  example, 
X  =  0,  y  =  0  is  evidently  a  solution  of  this  new  system  but  is  not  a 
solution  of  the  system  (1)  and  (2).     When,  however,  two  equations 

of   the   form     I  ^^  +    ^  —  ^^V  I     are   given,    one    solution    of    the 
\mx-\-  ny  =  pxy  J 


220 


ELEMENTARY  ALGEBRA 


system  may,  in  general,  be  obtained  by  dividing  each  member  of 
both  equations  by  xy  and  proceeding  as  in  the  solution  of  the  fore- 
going illustrative  example. 

+  -^  =  12, 


2.    Solve  the  system 


x-1 
5 


^  +  2 
3 


1. 


x-1      y+2 
Solution.     Writing  (1)  and  (2)  in  another  form, 


(1) 

(2) 

(8) 
(4) 


In  equations  (3) and  (4)  we  may  regard  the  unknown  numbers  as 
and  „.     Multiplying  both  members  of  (3)  by  3  and  both 


X  -I  y  -\-2 

members  of  (4)  by  2,  and  adding 


19 


{-^^- 


38. 


Dividing  by  19, 


=    2. 


x-1 

Clearing  (6)  of  fractions,  1  =  2(a;  -  1). 

Solving  (7),  ar  =  f . 

Substituting  in  (1)  the  value  of  x  as  found  in  (8), 


6  + 


=  12. 


y  +  2 

Solving  (9),  y  =  -  f  • 

Therefore,  the  solution  of  (1)  and  (2)  is  x  =  \,  y  =  - 
Check.     Substituting  in  (1)  and  (2), 


(5) 

(6) 

(7) 
(8) 

(») 
(10) 


-1      -f+2 
5  3 


=  12, 
=  L 


Simplifying, 


1      -I+.2 

I    6  +  6  =  12, 
1 10  -  9  =  L 


SYSTEMS  OF  LINEAR  EQUATIONS 


221 


EXERCISE  80 

Solve  the  following  systems  of  equations,  regarding  each 
as  a  system  of  simultaneous  linear  equations  in  two  un- 
knowns : 


5. 


3, 


?4-l 
X     y 

1  +  1  =  2. 

X     y 

X  y 
3_2 
X     y 

7 


=  2. 


x—\      y+b 


=  11, 


x-1      y+B 

17 

■  2 

2a     5a     3 
5  a     2  a      17 

2. 


4. 


6. 


8. 


'^+^=-1, 

X     y 

?-?=-4. 
X     y 


+ 


x+l      ?/  +  l 

1 


=  5, 


ah 
x  y 
a-\-  h      a 

X 


2, 
h 


=  1. 


a      h 

y     ~h      a 


y      20 


3a     5a^7 
hx       cy      4 ' 


7a_3« 
6a:       c^ 


23 

20 


10. 


11. 


f  2(a  +  6)  ,  3(a-6)  _  3 
X  y  2 

-3(^4-6)      5(a- 6)  ^11 
X  y  12" 

5       3a^a-18 
6x     5  y         6 
2       3a^2a  +  45 
3a:     5^/  15      ' 

a  _b 
X  y 
a      h 


=  m, 


=  n. 


y 


222 


ELEMENTARY  ALGEBRA 


12. 


13. 


14. 


h a_ 

4  ax     4:hy 


2b 


3  ax 


a 
hy 


a—  c 
x^h 

a  +  c 


a  —  h 


=  0, 


x—h 
1 


y-c 
a  +  h  ^      2a(c-b') 
y—c     (a—  b){a  —  c} 
1  2 


x  +  b-\-  c 
2a-\-b-\-c 


2b-\-  c-\-a        a—b 


x+b-\-e        y-\-c  +  a       a-\-b  +  c 

167.  Fractional  equations.  In  the  case  of  systems  of 
simultaneous  fractional  equations  which  are  not  included 
among  those  considered  in  section  166,  it  is  usually  best 
to  clear  the  equations  of  fractions. 


Solve  the  system 


ILLUSTRATIVE  EXAMPLE 

2x-\-5y 
x 

4:X 


=  1. 


2 

1^  =  2. 


(1) 

(2) 


x  +  y 

Solution.  Clearing  equations  (1)  and  (2)  of  fractions  and  com- 
bining like  terms, 

x^by  =  2,  (3) 

2  ar  -  5  y  =  0.  (4) 

Adding  (3)  and  (4),  3  a:  =  2.  (5) 

Solving  (5),  X  =  2 .  (6) 

Substituting  in  (3)  the  value  of  z  as  found  in  (5), 

.V  =  y  =  T^-  (7) 

Therefore,  the  solution  of  (1)  and  (2)  is  j:  =  J,  y  =  ^.  These 
values  of  x  and  y  are  found  to  satisfy  the  given  equations  (1)  and  (2). 

Remark.  Before  clearing  an  equation  of  fractions,  each  fraction 
should  be  expressed  in  its  lowest  terms  (see  also  note,  page  193). 


SYSTEMS  OF  LINEAR  EQUATIONS 


223 


EXERCISE   81 

Solve  the  following  systems  of  equations  and  check  the 
results  ; 


1.     i 


3. 


7. 


11. 


£±1=3, 

X—  2 

x-hy 

y-2-y-r 


x  +  1 

y-1 


=  1. 


2y  +  3^3y  +  2 
2a:4-3      32^  +  2' 

a:  +  ^  +  2  =  0. 

1    .   a;_2^-3a: 
1  -l —  —  —  , 

y        y 

Zx^y=l. 


x-\-2      y-3' 

a  b 

a  _h 
y^  x' 
2a         1 


ax-\-by      b 


2. 


4. 


8. 


10. 


12. 


=  3, 


y 

X 


x  + 


=  4, 


3y-2 

^y    2 

2a;-3y^2 
3a:-2?/      3' 
3a;H-2y_3 
2x-^y      2* 

a;  4-  ^  _  53      13  a;  4-  5 
3  2:    "39         26  a;    " 

a:+l  a;+l 

a:  -f  2  _  ,y  —  6 
a  a—  6' 


x—  y  =  a  —  b^ 

a  H-  c_  6+  c 
a:  ^ 


EXERCISE   82 

1.    The  sum  of  two  numbers  is  15  and  one  of  them  is 
one  greater  than  the  other.     What  are  the  numbers  ? 


224  ELEMENTARY  ALGEBRA 

Suggestion.     Let  x  =  the  larger  number  and  y  =  the  smaller. 
Then,  p  +  y  =  15,  (1) 

1         ^  =  ^  +  1.  (2) 

2.  The  sum  of  two  numbers  is  12,  and  their  difference 
is  6.     What  are  the  numbers  ? 

3.  The  sum  of  two  numbers  is  27,  and  five  times  the 
first  number  is  equal  to  four  times  the  second.  What  are 
the  numbers  ? 

4.  The  difference  between  two  numbers  is  5,  and  the 
sum  of  the  numbers  is  twice  their  difference.  Find  the 
numbers. 

5.  Twice  a  certain  number  is  4  greater  than  5  times 
a  second  number  ;  the  sum  of  the  two  numbers  is  80. 
Find  the  numbers. 

6.  A  bushel  of  corn  and  a  bushel  of  oats  together 
weigh  88  lb.,  and  the  weight  of  a  bushel  of  corn  is  24  lb. 
greater  than  the  weight  of  a  bushel  of  oats.  What  is  the 
weight  of  a  bushel  of  each  ? 

7.  The  weight  of  3  bu.  of  bran  is  equal  to  the  weight 
of  1  bu.  of  wheat,  and  the  weight  of  1  bu.  of  wheat 
exceeds  the  weight  of  1  bu.  of  bran  by  40  lb.  What  is 
the  weight  of  a  bushel  of  each  ? 

8.  The  sum  of  two  numbers  is  18,  and  4  times  the 
larger  is  equal  to  5  times  the  smaller.  What  are  the 
numbers  ? 

9.  Three  times  a  certain  number  is  7  greater  than 
four  times  the  sum  of  8  and  a  second  number  ;  the  sum 
of  three  times  the  first  number  and  four  times  the  second 
is  63.     Find  the  numbers. 

10.  One  half  of  one  number  is  equal  to  two  thirds  of 
a  second;  the  sum  of  the  first  number  and  twice  the 
second  is  20.     What  are  the  numbers  ? 


SYSTEMS  OF  LINEAR  EQUATIONS  225 

11.  A  classroom  has  54  desks,  some  of  which  are  single 
and  some  double  ;  the  seating  capacity  of  the  room  is  72. 
How  many  desks  of  each  kind  are  there  ? 

12.  Two  opposite  numbers  which  differ  by  8  have  the 
same  absolute  values.     What  are  the  numbers  ? 

13.  2  lb.  of  coffee  and  6  lb.  of  sugar  cost  11.18;  51b. 
of  coffee  and  3  lb.  of  sugar  cost  f  1.99.  Find  the  cost  of 
a  pound  of  each. 

14.  If  12  gallons  of  milk  will  just  fill  either  152  bottles 
and  5  jars,  or  32  bottles  and  20  jars,  what  are  the  separate 
capacities  of  a  bottle  and  a  jar  ? 

15.  A  dealer  bought  30  bu.  of  wheat  and  10  bu.  of  rye 
for  $46.  He  also  bought  at  the  same  time  50  bu.  of  wheat 
and  30  bu.  of  rye  for  $87.  Find  the  price  of  each  per 
bushel. 

16.  The  sum  of  two  numbers  is  equal  to  5,5  diminished 
by  the  second  number ;  three  times  the  first  number  dimin- 
ished by  twice  the  second  number  is  —  1.1.  What  are 
the  numbers? 

17.  "  Give  me  five  of  your  marbles,"  said  a  boy  to  his 
brother,  "and  I  shall  have  twice  as  many  as  you."  His 
brother  replied,  ^  Give  me  five  of  your  marbles  and  then 
I  shall  have  as  many  as  you."  How  many  marbles  had 
each  ? 

18.  Three  years  ago  a  boy  was  twice  as  old  as  his  sister, 
and  fifteen  years  hence  |  of  his  age  will  equal  J  of  his  sis- 
ter's age.     How  old  is  each? 

19.  A  bill  amounting  to  $8.70  was  paid  with  60  coins, 
some  of  which  were  dimes  and  the  rest  quarters;  how 
many  of  each  were  there  ? 


226  ELEMENTARY  ALGEBRA 

20.  Divide  $10  between  A  and  B,  so  that  the  number 
of  half-dollars  in  A's  share  may  be  ten  less  than  the  num- 
ber of  quarter-dollars  in  B's  share. 

21.  A  certain  number  is  equal  to  seven  times  the  sum 
of  its  two  digits,  and  the  left-hand  digit  exceeds  the  right- 
hand  digit  by  2.    Find  the  number. 

Suggestion.     Let         x  =  the  tens*  digit, 
and  y  =  the  units'  digit. 

Then,  \^  x  ■\-  y  =  the  number. 

Whence,  f  10  x  +  ^^  =  7(:r  +  y),  (1) 

\       x-y^%  (2) 

22.  The  length  of  a  room  is  25  %  greater  than  the  width, 
and  the  perimeter  is  35  ft.     Find  the  dimensions. 

23.  A  merchant  has  tea  worth  50  cents  per  pound  and 
also  tea  worth  70  cents  per  pound ;  how  many  pounds 
of  each  must  he  use  to  make  a  mixture  of  25  pounds  worth 
62  cents  per  pound  ? 

24.  The  cost  of  sending  a  day  telegram  of  17  words 
from  Philadelphia  to  Richmond,  Indiana,  is  71  cents,  and 
the  cost  of  sending  one  of  23  words  is  89  cents.  What  is 
the  rate  for  the  first  ten  words  in  such  a  message  and  for 
each  additional  word? 

25.  Five  first-class  fitters  and  7  plain  sewers  earn  $160 
a  week;  7  first-class  fitters  and  2  plain  sewers  earn  il85 
a  week.  Find  the  weekly  wages  of  a  first-class  fitter  and 
those  of  a  plain  sewer. 

26.  The  sum  of  two  digits  of  a  certain  n.umber  is  10 
and  if  18  be  added  to  the  number,  its  digits  will  change 
places.     Required  the  number. 

Suggestion.  Let       x  —  the  tens'  digit, 
and  y  =  the  units'  digit. 

Then,  10  a;  +  y  =  the  number, 

and  10  y  +  a:  =  the  number  with  the  digits  interchanged. 


SYSTEMS  OF  LINEAR  EQUATIONS  227 

27.  One  digit  is  one  greater  than  twice  a  second  digit ; 
the  difference  between  the  numbers  which  can  be  repre- 
sented by  the  two  digits  is  45.     P'ind  the  digits. 

28.  A  man  rode  a  certain  distance,  at  a  uniform  rate, 
in  7  hr.  If  the  distance  had  been  4  miles  less  and  his  rate 
per  hour  1  mile  more,  the  time  required  would  have  been 
6  hr.     Find  the  distance  and  his  rate. 

29.  One  man  and  three  boys  can  do  a  piece  of  work  in 
2f  working  days  of  10  hours  each ;  two  men  and  one  boy 
could  do  it  in  the  same  time.  How  many  hours  would 
one  man  alone  require  to  do  the  work? 

Suggestion. 

Let  X  =  the  number  of  hours  in  which  a  man  can  do  the  work, 

and  y  =  the  number  of  hours  in  which  a  boy  can  do  the  work. 

Then,      -  =  the  part  of  the  work  the  man  does  in  1  hr., 

X 

and  -  =  the  part  of  the  work  a  boy  does  in  1  hr. 

y 

Whence, 

l  +  ?  =  i-  (1) 

X     y     24  ^^ 

?  +  i  =  i-.  (2) 

X      y     2^  •  ^  ^  ^ 

30.  One  man  and  two  boys  can  do  a  piece  of  work  in 
9  days;  two  men  and  five  boys  could  do  it  in  4  days. 
How  long  would  one  boy  alone  take  to  do  the  work  ? 

31.  If  1  is  added  to  the  numerator  of  a  fraction,  the 
value  of  the  fraction  becomes  |- ;  if  1  is  added  to  the  de- 
nominator of  the  same  fraction,  the  value  becomes  J.  What 
i^  the  fraction  ? 

Suggestion.     Let  -  =  the  fraction. 


228  ELEMENTARY  ALGEBRA 

32.  If  1  be  added  to  both  terms  of  a  fraction  the  re- 
sulting fraction  will  be  |,  but  if  1  be  subtracted  from  both 
terms,  the  resulting  fraction  will  be  ^,  What  is  the 
fraction  ? 

33.  Separate  53  into  two  parts  such  that  the  greater 
part  divided  by  the  less  shall  give  both  a  quotient  and  a 
remainder  of  2. 

34.  A  owes  1250  and  B  owes  $375.  A  could  pay  all 
his  debts  if  in  addition  to  his  own  money  he  had  ^  of  B's ; 
and  B  could  pay  all  of  his  debts  and  have  $25  left  if  in 
addition  to  his  own  money  he  had  |  of  A's.  How  much 
money  has  each? 

35.  The  base  of  a  rectangle  is  10%  greater  than  the 
altitude,  and  the  perimeter  is  126  ft.     Find  the  dimensions. 

36.  A  part  of  $3000  is  invested  at  5|^%  and  the  remain- 
der at  4^%.  The  yearly  income  from  the  investments  is 
$147.25.     Find  the  amount  in  each  investment. 

37.  In  a  certain  family  each  son  has  twice  as  many  sis- 
ters as  brothers  but  each  daughter  has  as  many  brothers 
as  sisters.     How  many  children  are  in  the  family? 

38.  In  a  certain  family  each  daughter  has  as  many 
brothers  as  sisters,  but  each  son  has  three  times  as  many 
sisters  as  brothers.     How  many  children  are  in  the  family? 

39.  A  man  has  $7000  which  he  wishes  to  invest  in  two 
enterprises  so  that  his  total  income  Avill  be  $  330 ;  if  one 
enterprise  pays  5  %  and  the  other  4|  %,  how  much  must  he 
invest  in  each? 

40.  A  certain  principal  will  amount  to  $260  if  loaned 
at  simple  interest  for  5  yr. ,  and  to  $  240  if  loaned  at  the 
same  rate  for  4  yr.     Required  the  principal  and  the  rate. 


SYSTEMS  OP  LINEAH  EQUATIONS  229 

41.  A  certain  principal  in  a  given  time  will  amount  to 
$744  if  loaned  at  simple  interest  at  6%,  and  to  $708  if 
loaned  for  the  same  time  at  4|  %.  Required  the  principal 
and  the  time. 

42.  In  an  athletic  meet  the  winning  team  scored  42 
points  and  the  second  team  35  points.  The  winning  team 
took  first  place  in  6  events  and  second  place  in  4 ;  the  sec- 
ond team  took  4  first  and  5  second  places.  How  many 
points  does  a  first  place  count  and  how  many  does  a  second 
place  count? 

168.  Simultaneous  linear  equations  in  three  unknown 
numbers.  Three  consistent  linear  equations  in  three 
unknown  numbers  have  one  and  only  one  solution  when- 
ever by  elimination  two  independent  and  consistent  linear 
equations  in  two  unknowns  can  be  derived  from  them. 

ILLUSTRATIVE,  EXAMPLES 

2;  4-2^/4-3  2  =  4,  (1) 

2^  +  3^/4-42  =  7,  (2) 

[^x--ii/-5z  =  S.  (3) 

Solution.     Multiplying  (1)  by  2,     2  x  +  4  ?/  +  6  z  =  8.  (4) 

Subtracting  (2)  from  (4),  y  -h2z  =  l.  (5) 

Multiplying  (1)  by  3,  3  a;  +  6  2^  +  9  2  =  12.  (6) 

Subtracting  (3)  from  (6),  10y-hUz  =  i.  (7) 

Dividing  (7)  by  2,  5y-{-7z  =  2.  (8) 

Equations    (5)    and    (8)    are  two   independent   equations  in  two 

unknowns  and  are  solved  by  methods  previously  explained ;  thus : 
Multiplying  (5)  by  5,  5  3/  + 10  z  =  5.  (9) 

Subtracting  (8)  from  (9),   '  3  z  =  3.  (10) 

Solving  (10),  z  =  l.  (11) 

Substituting  in  (5)  the  value  of  2  as  found   in  (11)  and  solving 

resulting  equation  for  y,  ?/  =  —  !•  (12) 

Substituting  in  (1)  the  value  of   y  from  (12)  and  the  value  of  z 

from  (11),  and  solving  for  x,  x  =  3.  (13) 

Therefore,  the  solution  of  the  given  system  is  x=3,  y=  —1,  z  =  l. 


Solve  the  system 


230  ELEMENTARY  ALGEBRA 

Check.     Substituting  3  for  x,  —  1  for  //,  and  1  for  z  in  equations 
(1),  (2),  and  (3),  we  have  respectively, 


r  3  -  2  +  3  =  4. 

(14) 

.  6  -  3  +  4  :=  7. 

(15) 

9  4.  4  _  5  =  8. 

(16) 

?-?  +  l  =  i, 

(1) 

X      y      z 

4,23          2 

r                          -  5 

(2) 

X      y      z           3 

2      5      2_1^ 

(3) 

2.   Solve  the  system 


Solution.     Regard  equations  (1),  (2),  and  (3)  as  linear  in  the 

three  unknowns  -,-,-• 
X    y    z 

Multiplying  (1)  by  3,  §_?  +  ?=  3.  (4) 

X     y     z 

Adding  (2)  to  (4),  ---  =  ^  (5) 

X      y      Z 

Multiplying  (1)  by  2,  ?  _  i  +  ?  =  2.  (6) 

X      y      z 

Subtracting  (3)  from  (6),      '  ^  + 1  =  11.  (7) 

X      y       Q 

Multiplying  (7)  by  4,  16_^4^^  ^g^ 

X      y      6 

Adding  (8)  to  (5),  ??=f.  (9) 

X        6 

Solving  (9),  x  =  3.  (10) 

Substituting  in  (7)  the  value  of  x  as  found  in  (10), 

..      h]-l    ("> 

Solving  (11),  y  =  2.  (12) 

Substituting  in  (1)  the  value  of  x  from  (10)  and  the  value  of  y 

from  (12),  and  simplifying,  -  =  1.  (13) 

z 

Solving  (13),  z  =  \.  (14) 

Therefore,  the  solution  of  the  given  system  is  a;  =  3,  3/  =  2,  *  =  1. 


SYSTEMS  OF  LINEAR  EQUATIONS 


231 


-l  +  i 


1  =  - 
1 


Check.  Substituting  3  for  ar,  2  for  y,  and  1  for  2,  in  equations  (1), 
(2),  and  (3),  we  have,  respectively, 

-       -       '       1,  (15) 

(16) 
6-  (17) 

From  the  foregoing  illustrative  examples  we  may  infer 
the  following  : 

Rule.  To  solve  three  linear  equations  in  three  unknown  num- 
bers^ eliminate  any  one  of  the  unknowns^  as  x^from  any  pair 
of  the  equations^  and  then  eliminate  the  same  unknown  from 
another  pair  ;  solve  the  resulting  two  linear  equations  in  two 
unknowns  for  these  unknowns^  substitute  the  values  of  the  two 
unknowns  in  one  of  the  given  equations^  and  solve  for  the  third 
unknown  number. 

BXEBCISE  83 

Solve  the  following  systems  of  equations,  and  check  the 
results : 


1. 


3. 


7. 


2a?-3  «/-2=12,  2. 

3a:  +  ^+  2z  =  5. 
Sx+2y-lz=-14:, 
Sx-2y  +  5z  =  m,       4. 
x-{-1  y-2z=-29. 
llx+2y-{-Sz  =  24:, 
5a;H-3^-42=-18,       6. 
2x-5y  +  l  z  =  ^2. 
5x-Sy-\-2z  =  U, 
4a;  +  4^-3  2  =  57,       8. 
Sx-h2y-\-5z  =  16. 
x-\-y-\-z  =  2, 
2x-Sy-{-llz  =  S,     10. 
[Sx-^ly-2z=5, 


[2x-3^  +  5z  =  15, 

3a;+2y-4z=-7, 

x-\-  y-^z=:2, 
\5x-\-2y-\-Sz=4:, 

Sx-Sy-\-4z=-19, 

2x-\-3y-l  z  =  41. 

Sx-^2y+Sz  =  S, 

22:4-3^  +  22  =  27, 

1  x-5y-5z  =  91. 

5x-2y  =  S, 

3a;4-2z  =  5, 

By-Sz=2, 

x-\-  y=0, 

?/  +  2  =  -l, 

z-^x  =  l. 


232 


ELEMENTARY  ALGEBRA 


11. 


13. 


15. 


[1  .  1 

1_ 

-  +  -- 

6, 

X     y 

z 

1      1 

\  _ 

---  + 

-2, 

X     y 

z 

1      1 

1_ 

-4--  + 

0. 

.^    y 

z 

W-= 

5, 

X      y 

1      1 

-+-  = 

6, 

y      z 

1+1= 

7. 

Z        X 

12. 


14. 


11 

6"' 


f2_^3_l 
X     y     z 
3      1 ,2_7 
X     y     z      Q 

X     y     z      S 

1.1      o 

-  +  -  =  2  a, 

1  +  1  =  25, 

1^1      O 

-  +  -  =  2  (?. 

2         X 


{ ax -\- hy  +  cz  =  a^ -\- h'^  -\-  c^^ 

I  (5  +  c)x  +  (<?  4-  a)^/  +  (a  +  6)2  =  2  6c  +  2  m  4-  2  ah, 

\  hex  H-  m?/  +  ahz  =  3  a5<?. 


EXERCISE  84 

1.  I  paid  91  ct.  for  2  lb.  of  sugar,  1  lb.  of  coffee,  and  3 
lb.  of  lard.  If  I  had  bought  3  lb.  of  sugar,  1  lb.  of  coffee, 
and  21b.  of  lard,  my  bill  would  have  been  84  ct.;  but  if 
I  had  bought  1  lb.  of  sugar,  21b.  of  coffee,  and  1  lb.  of 
lard,  my  bill  would  ha-ve  been  83  ct.  What  did  I  pay 
for  a  pound  of  each  ? 

2.  For  16  1  can  buy  3  lb.  of  tea,  8  lb.  of  coffee,  and 
30  lb.  of  sugar  ;  or  4  lb.  of  tea,  7  lb.  of  coffee,  and  25  lb. 
of  sugar  ;  or  8  lb.  of  tea,  1  lb.  of  coffee,  and  15  lb.  of  sugar. 
What  are  the  prices  ? 

3.  There  are  three  numbers  whose  sum  is  162 ;  the 
second  exceeds  the  first  as  much  as  the  third  exceeds  the 
second ;  18  times  the  first  equals  5  times  the  third. 
What  are  the  numbers  ? 


SYSTEMS  OF  LINEAR  EQUATIONS  233 

4.  A  dealer  shipped  100  doz.  of  eggs  on  Monday  and 
Tuesday,  110  doz.  on  Tuesday  and  Wednesday,  and  90 
doz.  on  Monday  and  Wednesday.  How  many  dozen  did 
he  ship  each  day  ? 

5.  A  man  has  a  triangular  lot  which  he  desires  to 
fence.  100  rods  of  fencing  are  required  for  the  sides  AC 
and  BC,  111  rods  for  the  sides  AB  and  BC,  and  90  rods 
for  the  sides  AB  and  AC.  Find  the  number  of  rods  of 
fencing  required  for  each  side. 

6.  The  sides  of  a  certain  triangle  are  denoted  by  a,  6, 
and  c.  What  is  the  length  of  each  side  of  the  triangle  if 
the  sum  of  the  sides  is  42,  the  sum  of  a  and  h  is  27,  and 
the  sum  of  h  and  c  is  29  ? 

7.  The  sum  of  the  three  angles  of  any  plane  triangle 
is  180°.  If  these  angles  are  denoted  by  A^  B,  and  (7,  and 
if  O  exceeds  A  by  50°  and  A  exceeds  B  by  10°,  find  the 
size  of  each  angle. 

8.  The  sum  of  three  numbers  is  24.  The  quotient  of 
the  first  divided  by  the  second  is  |  and  the  quotient  of  the 
second  divided  by  the  third  is  |.     Find  the  numbers. 

9.  The  middle  digit  of  a  given  three-digit  number  is 
equal  to  the  sum  of  the  two  remaining  digits  ;  a  second 
number  which  is  594  less  than  the  given  number  is 
expressed  by  the  same  digits  written  in  the  reverse  order ; 
if  19  be  added  to  the  original  number  and  13  be  added  to 
the  second,  one  of  the  resulting  numbers  will  be  four 
times  the  other.     Find  the  original  number. 

10.  I  made  three  shipments  of  goods  from  Philadelphia 
to  Chicago.  The  first  consisted  of  300  lb.  first-class 
freight,  200  lb.  second-class  freight,  and  200  lb.  third- 
class  freight.  My  freight  bill  was  $4.65.  The  second 
shipment  consisted  of  1000  lb.  first-class  freight,  500  lb. 


234  ELEMENTARY  ALGEBRA 

of  second-class,  and  100  lb.  of  third-class  freight.  My 
freight  bill  was  111.51.  The  third  shipment  consisted  of 
700  lb.  of  first-class  freight,  800  lb.  of  second-class  freight, 
and  400  lb.  of  third.  My  freight  bill  was  $12.71.  Find 
the  rate  of  shipping  100  lb.  of  freight  of  each  class  from 
Philadelphia  to  Chicago. 

11.  A  and  B  together  can  do  a  piece  of  work  in  20 
days.  After  they  have  worked  12  days  on  it,  they  are 
joined  by  C,  who  works  twice  as  fast  as  B.  The  three 
finish  the  work  in  4  days.  How  long  would  it  take  each 
man  alone  to  do  it  ? 

12.  A  number  is  composed  of  three  digits,  whose  sum 
is  12  ;  the  digit  in  the  hundreds'  place  is  one  greater 
than  that  in  the  tens'  place  ;  if  ten  times  the  units'  digit  is 
subtracted  from  the  number  the  remainder  is  257.  Find 
the  number. 

EXERCISE  85.  — GENERAL  REVIEW 

(Solve  as  many  as  possible  at  sight.) 

1.  Adda;+  2^-1-3  2,  1x—y  —  ^z^  y  —  x  —  z^  and  z  —  x—y. 

2.  Add  4  m^  —  3  mn—  2  mhi  —  n^  -\-  ^  mn^^  2  mhi  —  4  mn^ 

—  m^-\-Sn^^   4  mh^  —  3  m^  -|-  4  mn  —  m'n?^    and    5  mn^  —2n^ 

—  2  m^n. 

3.  What  must  be  added  to  x-\-  y  -\-  z  that  the  sum  may 
be  c  —  zl 

4.  What  must  be  subtracted  from  x'^  —  x-\-l  that  the 
difference  may  be  a^  —  1  ? 

5.  Subtract  2(r  —  «)  -h  1  from  the  sum  of  7(r  —  «)  and 
-4(r-«). 

6.  Simplify 
2-(-2)-[-(-2)]-[-!-(-2)i-2]. 


SYSTEMS  OF  LINEAR  EQUATIONS  235 

7.  Express  m  —  n  —  p—q  +  r—s  —  t  —  u  —  x  in  trino- 
mial terms  having  the  last  two  terms  of  each  inclosed  in 
parentheses. 

If    A  =  E^-Rr  +  r\    ^  =  2i2H- r-f- 1,    and    C=4R^ 
—  2  Br —  2,  find  the  value  of : 

8.  A  +  B-^a         9.    A-B+O.        10.    A-B-C. 

11.  What  is  the  product  of  mVp  and  2  m^pq  ? 

12.  What  is  the  product  of  a:"-i«/"+%"  and  xy^~^z  ? 

13.  What  is  the  product  of  2(a:  +  ^),  Z(x-\-y)\  and 

14.  Divide  6  m^n^p^r  by  2  m^n^pr.  . 

15.  Divide  6a:;"+V^""^^  by  3  2:"y»-V+2. 

16.  Divide  6(m  +  7i)2  — 4(m-|-^)^+.(w  +  7i)  by2(m-hw). 

17.  Multiply  a:2  —  2  a;^  4-  «/^  -f  ^2  by  a;2  +  2  a:?/  +  ^2  _  ^2. 

18.  Multiply  a:*"  +  ^p  —  2  2"  by  2  a;"»  —  3  «^. 

19.  Multiply  f  :r2  +  f  a^y  4- i  ^2  by  I  a;2_  1:^:^4- 1^2. 

20.  Multiply  3"»  —  4"  by  4^  -h  3". 

21.  Divide 

24  mVpif  —  36  m^n^pT^  +  48  mn^r^x  by  —  6  ww2. 

22.  Divide  Qi?-^y^-\-^—^xyzh^x^^-y^-\-z^  —  yz  —  xz  —  xy, 

23.  Divide  1  -  547  :r«  +  546  a:^  by  1  +  2  a:  -  3  2^2. 

24.  Divide    3-5:^-487:^5-1-489  2:6    by    l-4a;  +  32^ 
and  find  the  value  of  the  quotient  when  rr  =  —  1. 

1  7^ 

25.  Show  by  division  that  -—■ —  =  \-\-x-\'Q^-{-^-\- . 

\—x  Y—x 

1  x^ 

26.  Show  by  division  that    =  1  —  rr-ha;2— a^S-f- 


\-\-  X  1  —  x 

27.    When  the  divisor  is  m^  +  n^  and  the  quotient  is 
m^  —  mV  H-  w%*  —  mV  H-  w^,  what  is  the  dividend  ? 


236  ELEMENTARY  ALGEBRA 

28.  When  the  divisor  is  m  —  n^  the  quotient  m^  +  rn^n 
_j_  y^2^2  ^  ^^3  ^  ^4^  an(j  ^]^g  remainder  2  7i^,  what  is  the 
dividend  ? 

29.  Simplify  4w  — {2  w—  \2n(r-\-  «)—  2  7i(r  —  «)]  j, 

30.  Divide  aP  —  y^  hy  x^  +  y^  -\-  a^y  -f-  xy^  +  a;^^^^ 

31.  Divide  (1  —  m^)  by  [(1  —  m)(l  +  w)]. 

32.  Divide  1  +  a^  4.  ^4  ]3y  j  ^  ^  ^  ^2^ 

33.  From  (2m  -\-n  ■\-p)x  take  (m  4-  7i)x. 

34.  From  (r  +  s)y  +  («  +  0^^  take  (r  —  «)?/  —  («  —  ^)z. 

35.  Square  as  indicated  : 

(2«+5)2;     (1-2^)2;     (l-fm)2;     (2  m  -  J)2. 

36.  Square  as  indicated :    (m^n—p)^;   (mn—ph'^')^, 

37.  (a  +  5)(a-6)(«2+62)^? 

38.  (m-l)(m  +  J)=? 

-  (i-5)(>-i)=' 

40.  Express  26  x  24  in  the  form  of  (a  +  b)(a—  h)  and 
state  the  product. 

41.  21x19=?    51x49  =  ?    53x47  =  ?    101x99  =  ? 

42.  Does(-^)(+2:)(-y)(+2)=<-2:)(+^)(-z)? 

Why? 

43.  Find  in  the  shortest  way  the  value  of  748  x  680 
-  748  X  670. 

44.  Find  in  the  shortest  way  the  value  of  2  TrMff-}-  2  ttR^ 
when  TT  =  3.1416,  i2  =  1,  and  iT  =  9. 

45.  What  must  be  added  to  x^  +  4:X  that  the  sum  may 
be  (x  4-  2)2  ? 

46.  What  must  be  added  to  a:^ -far  that  the  sum  may  be 


SYSTEMS  OF  LINEAR  EQUATIONS  237 

47.    What  must  be  subtracted  from  x^-^^xy-^y'^  that 
the  difference  may  be  {x—yYl 

Factor : 


48. 

m^  +  m3. 

49. 

K^-^)-H^-s^) 

50. 

r(a-h)-8{h- 

-a). 

51. 

a2-b^-2bc-c\ 

52. 

mx-\-ny  —  nx- 

my. 

53. 

64  -  m^ 

54. 

l^a-a^-haK 

55. 

l-x^-x  +  x\ 

56. 

m2 

57. 

X      x^ 

58. 

m^      n^ 
n^      m^ 

59. 

y.3 

60.    R^ -h Eh-^ -\- r^,  61.    4x^-^x-l. 

62.  Find  all  the   factors  of  Sx^-\-a^^l  x^- lOx- S, 
being  given  that  x^-\-x-\-l  is  one  of  them . 

63.  Find  all  the  factors  of  x^  -\-  x^  —  1  x^  —  x  +  6,  being 
given  that  two  of  them  are  x  —  2  and  x  —  1. 

64.  Reduce  to  lowest  terms    ^  ~  ^    » 

ix-  yy 

1  —(r  —  8^^ 

65.  Reduce  to  lowest  terms  ^^ ^  • 

8  —  rs  -\-  8^ 

66.  Simplify  2-^"^ ~^. 

Z-^y-X 

67.  Simplify  -1- + -J— 1-. 

^—y    y  —X     ^-\-  y 

68. 


69.    Simplify  — — ^ -^— —  x     ,        ,  < 

TTjii  —  77171  -\-n^      mP  —  w 


(x  +  iy 


7^-1 


238 


70.    Simplify 


ELEMENTARY  ALGEBRA 


x^  —  y"^ 


{m  +  n)^     m^  4-  3  n^m  +  3  nm^  -f-  n^ 


71.    Find  the  value  of  when  x  = 


n—  X  m-\-n 

72.  Given  a  =  jt?  +  'prt ;  find  f  in  terms  of  a,  r  and  t» 

»o     c  1       1,111 

73.  Solve  — +  -  = 

m     X     n     X 

74.  Solve  0.3  a: -0.1  4-^'^ '^"^•^  =  0.4  a: -0.05. 


1.2 


75.    Solve 


76.    Solve 


77.    Solve  . 


1  1 


=  -1. 


X     y 
3 


x-{-y—\      x-^y-\'2 
4  2 


=  0, 


=  0. 


78.    Solve 


a;— 8      3/  — 4 

x-{-y-\-z  =  S, 
4:x-Sy-\-2z  =  -2y 
6x-2y-Sz  =  l, 

X     y     z 


CHAPTER   VIII 

RATIO,   PROPORTION,   AND   VARIATION 

Ratio 

169.  Definition.     The  ratio  of  a  number  a  to  a  number  b 

is  the  quotient  -  obtained  by  dividing  a  by  b.     The  ratio 

a  to  6  is  sometimes  written  a  :  b. 

Note.  By  the  ratio  of  one  quantity  to  a  second  quantity  is  meant 
the  number  of  times  that  the  first  contains  the  second ;  as  the  ratio 
of  4  quarts  to  3  quarts  is  ^.  Obviously,  no  ratio  exists  between 
quantities  which  are  not  of  the  saine  kindy  and  before  the  ratio  of  two 
quantities  which  are  of  the  same  kind  can  be  found,  they  must  be  ex- 
pressed in  terms  of  the  same  unit. 

Thus,  the  ratio  of  one  gallon  to  three  quarts  is  the  ratio  of  four 
quarts  to  three  quarts,  which  is  ^. 

170.  Definitions.     In  the  ratio  f  the  dividend,  or  nu- 

b 

merator,  a,  is  called  the  first  term  or  antecedent,  and  the 
divisor,  or  denominator,  6,  is  called  the  second  term  or 
consequent  of  the  ratio. 

EXERCISE  86 

In  examples  1-24  express  the  ratios  as  fractions  and 
simplify  when  possible  : 

1.    2:4.  2.    6:8.  3.    9  :  3. 


4.    15:. 10. 

5.    a2  .  ^^ 

6.    aP:2^, 

7.    J:i. 

8.    i:|. 
239 

9.    ab^  :  b. 

240  ELEMENTARY  ALGEBHA 


10. 

ax\  Q^. 

11. 

n  :  5f . 

12. 

{x-y):i^- 

f\ 

13. 

Cx-\-i/y:(x-^y). 

14. 

9  a2^  :  15  a2^J. 

15. 

3  pk.  :  8  pk. 

16. 

2  rd.  :  161  rd. 

17. 

9  in.  :  1  ft. 

18. 

2  yd.  :  1  rd. 

19. 

5  pk. : 2  bu. 

20. 

5  gal.  :  3  qt. 

21. 

1  mill.  :  1  hr. 

22. 

ahc  :  - . 
c 

23. 

$x  :  y  ct. 

24     (a  -  5)  yd.  :  {a  +  5)  ft. 

•25.  Arrange  the  ratios,  |,  |,  J,  |,  in  ascending  order  of 
magnitude. 

26.  What  number  must  be  added  to  both  terms  of  the 
ratio  1^  in  order  to  convert  it  into  the  ratio  ^? 

27.  Two  numbers  are  in  the  ratio  of  7  to  4 ;  if  3  be 
subtracted  from  each  number,  the  differences  are  in  the 
ratio  of  11  to  5.     Find  the  numbers. 


Proportion 

171.  Definitions.  Four  numbers  or  quantities  are  said 
to  be  in  proportion  when  the  ratio  of  the  first  to  the  sec- 
ond is  equal  to  the  ratio  of  the  third  to  the  fourth. 

Remark.  In  what  follows  we  shall  use  the  expression,  the  ratio 
of  one  number  to  another  instead  of,  the  ratio  of  one  number  or  quantity 
to  another. 

If  Y  =  -,  then  a,  5,  <?,  d  are  in  proportion.     A  proportion 
0      a 


is,  therefore,  an  expressed  equality  of  two  ratios.     In 

proportion,  as  7  =  -  ,   the  n 
0      a 

the  terms  of  the  proportion. 


proportion,  as  7=^5   the  numbers  a,  6,   c,  d  are  called 
0     d 


.   RATIO,  PROPORTION,  AND  VARIATION         241 

Note.  A  proportion  is  often  written  a:b  =  c:d;  read,  the  ratio 
of  a  to  b  is  equal  to  the  ratio  of  c  to  d.  An  old  form  and  one  less 
frequently  used  is  a:b  :  :c  :d. 

172.  Extremes  and  means.  The  first  and  fourth  terms 
of  a  proportion  are  called  the  extremes  and  the  second 
and  third  terms  the  means. 

Thus,  in  the  proportion  -  ==  -,  a  and  d  are  the  extremes  and  b  and 
b    '  d 
c  the  means. 

173.  Important  identities.  If  the  four  numbers,  a,  b,  c, 
and  c?,  are  the  four  terms  of  a  proportion,  they  satisfy  the 
identity,  ^^^ 

Multiplying  both  members  of  identity  (1^  by  bd^  we 
^a^^'  ad=bc.  (I) 

Identity  (I)  may  be  expressed  in  words  as  follows  : 
In  any  'proportion  the  product  of  the  extremes  is  equal  to 
the  product  of  the  means. 

Again,  if  -t=-7  5  ^^^^  the  reciprocal  of  -  is  equal  to  the 
b     d  b 

reciprocal  of  - ;  that  is, 
d 

-  =  -.  (II) 

a     c 

Since  the  reciprocal  of  a  fraction  is  obtained  by  invert- 
ing the  fraction,  identity  (II)  is  sometimes  expressed  as 
follows  : 

1^ four  numbers  are  in  proportion^  they  are  also  in  pro- 
portion by  inversion. 

If  7  =  ;^>  WG  have  from  identity  (I), 


ad  =  be.  (2) 


242  ELEMENTARY  ALGEBRA 

Dividing  both  members  of  identity  (2)  by  ah^ 

{='-  (HI) 

o     a 
Again,  dividing  both  members  of  identity  (2)  by  dc^ 

c     a 

Comparing  identities  (III)  and  (IV)  with  the  propor- 
tion ^  =  - ,  we  observe  that  if  four  numbers  taken  in  a 
h     d 

certain  order  are  in  proportion,  they  continue  to  be  in 
proportion  when  either  extremes  or  means  are  inter- 
changed.    This  fact  is  usually  expressed  as  follows  : 

If  four  numbers  are  in  proportion^  they  are  also  in  propor- 
tion hy  alternation. 

If  1  be  added  to  both  members  of  the  identity 

(3) 

(4) 

(V) 


a     c 
h^d' 

3  have, 

h             d 

Combining, 

a  +  b     c-h  d 

Subtracting  1  from  both  members  of  identity  (3), 

Combining,  ~     =     ""    .  (VI) 

0  d 

Dividing  the  members  of    (V)    by  the   corresponding 
members  of  (VI), 

fl-t-  fr^c  +  rf  rvin 

a-b    c-d'  ^    ■ 


RATIO,  PROPORTION,  AND  VARIATION         243 

Identities  (V),  (VI),  and  (VII)  are  usually  expressed 
in  order  as  follows: 

If  four  numbers  are  in  proportion^  they  are  also  in  propor- 
tion hy  composition. 

If  four  numbers  are  in  proportion^  they  are  also  in  pro- 
portion by  division. 

If  four  numbers  are  in  proportion^  they  are  also  in  propor- 
tion by  composition  and  division. 

EXERCISE  87 

Test  identities  (I)- (VII)  by  the  use  of  the  propor- 
tions of  examples  1-4. 


^      «  31     210 

a  —  b 
5a _  30  ab  ax-\-  ay  —  bx  ^  by  _a  -\-  b 

3  5      18  52  '    ax  —  ay  -\-  bx  —  by     x  —  y 


x-\-  y 
Find  the  value  of  x  in  each  of  the  proportions  stated  in 
examples  5-12. 

5.    1  =  -.  6.    5:  2  =  2::  10. 

3     X 

7.    3:  2:=  6:  14.  8.    2: :  10  =  5  :  2. 

9.    — i^=— ^t^.  10.    a-\-x:b-\-x=:c-{-x:d+x. 

S  -\-  X     lb  -\-  X 

11.    3-x:  -2  =  3a:+4:32.     12.    a:b::x:c. 

13.  Write  by  inversion ; 

yd  y      3  n      q 

14.  Write  (a),  (6),  and  (c)  of  example  13  by  alter- 
nation, 


244  ELEMENTARY  ALGEBRA 

15.  Write  (a),  (6),  and  (<?)  of  example  13  by  com- 
position. 

16.  Write  (a),  (6),  and  (c?)  of  example  13  by  division. 

17.  Write  (a),  (6),  and  (c)  of  example  13  by  compo- 
sition and  division. 

18.  If  four  numbers  are  proportionals  (in  proportion), 
prove  that  either  mean  is  equal  to  the  product  of  the 
extremes  divided  by  the  other  mean. 

19.  If  four  numbers  are  proportionals,  prove  that  either 
extreme  is  equal  to  the  product  of  the  means  divided  by 
the  other  extreme. 

20.  If  the  product  of  two  numbers  is  equal  to  the 
product  of  two  other  numbers,  prove  that  a  proportion 
may  be  formed  by  taking  one  pair  of  numbers  for  the 
extremes  and  the  other  pair  for  the  means. 

21.  Using  the  statement  of  example  20,  write  the  eight 
proportions  that  may  be  expressed  from  the  identity 
mq  —  np. 

22.  It  -  =  -,  prove  that  — ' —  =  — ' — . 

0      d  a  c 

Suggestion.  Write  the  given  proportion  by  inversion,  and  then 
apply  identity  (V). 

23.  It  7  =  -,  prove  that  = . 

0      d  a  c 

174.  Continued  proportion.  Three  or  more  numbers  are 
said  to  be  in  continued  proportion  when  the  ratio  of  the 
first  to  the  second  is  equal  to  the  ratio  of  the  second  to 
the  third,  and  so  on. 

Thus,  a,  6,  c,  d  are  in  continued  proportion  if, 
a:h  =  h  :c  =  c  :d. 


IIATIO,  PR0I>0RTI0N,  AND  VARIATION        245 

175.  Mean  proportional.  When  three  numbers  are  in 
continued  proportion,  the  second  is  said  to  be  a  mean  pro- 
portional between  the  other  two. 

Thus,  if  -  =  -,  the  number  &  is  a  mean  proportional  between  the 
b      c 

extremes  a  and  c. 

176.  Third  proportional.  When  three  numbers  are  in 
continued  proportion,  the  third  is  said  to  be  a  third  pro- 
portional to  the  other  two. 

Thus,  if  -  =  -,  the  number  c  is  a  third  proportional  to  a  and  b. 
b      c 

VJl.  Fourth  proportional.  A  fourth  proportional  to 
three  numbers  a,  5,  e,  taken  in  the  order  given  is  the 
fourth  term  of  the  proportion 

a  :  h  =  c  :  X. 

178.  Composition  of  equal  ratios.     Let  -,  -,  and  -  be 

ha  J 

equal  ratios  and  each  equal  to  r ;  that  is. 


ace 

Then, 

a  —  hr^  c=  dr,  e  —fr. 

Adding, 

a  +  c+^  =  (^  +  c?  -h/)r. 

Dividing, 

«  +  <?+«_ 

Therefore, 

a-fc  +  e     a     c     e 
h+d+f     b     d    f 

This  identity  may  be  expressed  in  words  as  follows  : 
In  a  number  of  equal  ratios  the  sum  of  the  antecedents  is 

to  the  sum  of  the  consequents   as  any  antecedent   is   to   its 

consequent. 


246  ELEMENTARY  ALGEBRA 

EXERCISE  88 

1.  If  a,  6,  and  c  are  three  numbers  in  continued  pro- 
portion, show  that  6^  =  ac. 

2.  If  a,  6,  and  c  are  thre^  numbers  in  continued  pro- 
portion, show  that  c,  the  third  proportional  to  a  and  6,  is 

equal  to  —  • 
a 

3.  What  positive  integer  is  a  mean  proportional  be- 
tween 256  and  36  ? 

4.  Express  by  proportion  the  fact  that  6  is  a  mean 
proportional  between  4  and  9. 

5.  Express  by  proportion  the  fact  that  18  is  the 
third  proportional  to  2  and  6. 

6.  What  is  the  third  proportional  to  10  and  5  ? 

12a 

7.  If  -  =  -  =  -,  what  are  the  values  of  a  and  h  ? 

2      a      J' 

8.  If  !^  =  £  =  !:,  show  that  ^?^+^±^  =  ^  =  ^  =  ^. 

n       q      8  n  -\-  q-\-  s       n      q      8 

9.  From  the  proportion  |  =  f ,  derive  another  propor- 
tion by  inversion  ;  derive  two  other  proportions  by  alter- 
nation ;  derive  another  proportion  by  inversion  and 
composition ;  derive  another  proportion  by  division ;  de- 
rive another  proportion  by  inversion  and  division. 

,^     y£  a      c  4.1    4.  «      ^      3a-f-  2  c 

10.  If  7  =  - ,  prove  that  -  =  -  =  —: —-_  • 

Suggestion.     Since  -  =  - ,  then  —  =  — ^  ;  now  see  section  178. 
h      d  3  6      2r/ 

11.  If  _^  = =  — ^ ,  and  a-\-h  +  c  is  not  equal 

h  -\-  e      c  +  a      a-\-  0 

to  zero,  prove  that  each  fraction  is  equal  to  J.  When 
a-\-  b-\-  c  is  equal  to  0,  what  is  the  value  of  each  of  the 
fractions  ? 


RATIO,  PROPORTION,  AND  VARIATION         247 

Variation 

179.  Constant.  A  number  which  has  always  the  same 
value  is  a  constant. 

Thus,  2,  —  ^,  and  a,  supposing  the  value  of  a  to  be  known,  or 
given,  are  constants ;  also  the  root  of  a  simple  numerical  equation  in 
one  unknown  number  is  a  constant ;  as  the  root  of  2  2-  —  4  =  0. 

180.  Variable.  Many  of  the  numbers  of  algebra  are 
not  constants.  A  number  which  is  not  a  constant  is 
called  a  variable. 

Thus,  if  the  case  of  an  express  train  running  from  New  York  to 
Philadelphia  be  considered,  and  i  denotes  the  number  of  minutes 
which  have  elapsed  since  it  left  New  York,  s  the  number  of  feet  that 
it  passed  over  in  t  minutes,  and  v  the  number  of  feet  that  it  passed 
over  in  each  minute,  v  supposed  to  be  constant,  then  the  formula 
connecting  the  time,  the  rate,  and  the  distance  passed  over  is, 

s  =  vt. 

In  this  formula  v  is  constant,  but  t  and  s  are  both  variables  during 
the  whole  time  that  the  train  is  in  motion. 

Remark.  The  unknown  numbers  in  an  equation  in  two  or  more 
unknowns  are  called  variables. 

Thus,  in  y  =  2  r,  any  value  whatsoever  may  be  assigned  to  either 
X  or  y.  The  fact  that  it  is  not  necessary  to  assign  one  fixed,  or  con- 
stant, value  to  X  or  y,  as  is  the  case  in  a  simple  equation  in  one  un- 
known, in  order  to  obtain  a  solution  of  the  equation,  is  sufficient 
reason  for  calling  x  and  y  variables  in  the  equation. 

181.  Application  of  terms.  In  general,  a  letter  is  said 
to  be  a  variable ;  that  is,  it  represents  a  variable  number 
if  it  may  have  a  number  of  different  numerical  values  in 
a  discussion  or  problem.  It  is  a  constant  if  it  can  have 
only  one  numerical  value. 

182.  Direct  variation.  If  one  variable,  y,  depends 
upon  another  variable,  x^  in  such  a  way  that  the  ratio  of 
^  to  a:  is  a  constant,  then  y  is  said  to  vary  as  x,  or  vary 
directly  as  x. 


248  ELEMENTARY  ALGEBRA 

If  y  varies  as  x^  then, 

y.=ic   (a  constant).  ^  ^ 

X 

X      1 
Hence,  -  =  -  (a  constant).  (2) 

y     0 

From  (1)  and  (2)  it  is  evident  that  if  y  varies  directly 
as  a:,  then  x  varies  directly  as  y, 

183.  Illustrations  of  direct  variation.  1.  The  number 
of  cents  (c)  in  the  cost  of  cloth  bought  at  the  fixed  price 
of  60  cents  per  yard  varies  as  the  number  of  yards  (n) 

bought,  since  the  ratio  —  is  a  constant,  namely,  the  num- 

n 

ber  of  cents  in  the  cost  of  one  yard. 

2.    In  the  formula  8  =  vt^  in  which  i;  is  a  constant,  « 

varies  directly  as  t ;  that  is,  the  space  passed  over  varies 

directly  as  the  time. 

Note.  In  the  equation  s  =  vt,  as  elsewhere,  s,  v,  and  t  represent 
numbers,  and  the  statement,  *^  space  passed  over  varies  directly  as  the 
time"  means  that  the  number  s  varies  as  the  number  /. 

184.  Inverse  variation.  If  the  variable  y  varies  as 
the  reciprocal  of  x;  that  is,  if  y  is  equal  to  a  constant 
times  the  reciprocal  of  x^  then  y  is  said  to  vary  inversely 
as  X. 

Thus,  \i  y  —-,  where  c  is  a  constant,  y  varies  inversely  as  x. 

X 

Also,  from  y  =-  we  have  x  =  -,  and  hence  x  also  varies  inversely 

X  y 

as  y ;  moreover,  from  y  z=  -  we  derive  xy  =  c ;  hence, 

X 

If  the  product  of  two  variables  is  a  constant^  the  one  varies 
inversely  as  the  other. 

185.  Illustration  of  inverse  variation.  The  number  of 
yards  of  cloth  (n)  that  can  be  bought  for  a  certain  fixed 


RATIO,  PROPORTION,  AND  VARIATION         249 

price,  as  $8,  varies  inversely  as  the  number  of  dollars  (r) 
that  the  cloth  costs  per  yard,  since  ti  =  - ;  that  is,  n  is  a 
constant  times  the  reciprocal  of  r. 


ILLUSTRATIVE   EXAMPLES 

1.  If  y  varies  directly  as  x^  and  y  is  100  when  x  is  20, 
what  equation  expresses  the  relation  between  y  and  a;? 

Solution.  Since  y  varies  as  x,  then  y  is  some  constant,  as  c, 
times  X ; 

Therefore,  y  =  ex.  (1) 

Since  y  is  equal  to  100,  when  x  is  equal  to  20, 

100  =  20  c.  (2) 

Solving  (2)  for  c,  c  =  5.  (3) 

Substituting  the  value  of  c  in  (1), 

y  =  5x.  (4) 

2.  The  number  of  feet  that  a  body  falls  from  rest  under 
the  action  of  gravity  is  proportional  to  (varies  directly  as) 
the  square  of  the  number  of  seconds  during  which  it  falls. 
If  a  body  falls  144  feet  in  three  seconds,  how  far  will  it 
fall  in  five  seconds?  How  many  seconds  are  required  for 
it  to  fall  192  yards? 

Solution.  Let  s  represent  the  number  of  feet  that  the  body  falls, 
and  t  the  number  of  seconds  during  which  it  falls ;  also,  let  c  repre- 
sent a  constant. 

Since  s  varies  as  t^,  s  =  cfi.  (1) 

Since  s  =  144,  when  <  =  3,     144  =  9  c.  (2) 

Solving  for  c,  c  =  16.  (3) 

Substituting  the  value  of  c  in  (1), 

s  =  16 1\  (4) 

To  find  how  far  the  body  will  fall  in  five  seconds,  substitute  5  for 
/  in  (4);  then,  s  =  16  x  25  =  400.  (6) 

Hence,  the  body  falls  400  feet  in  5  seconds. 

Also,  to  find  the  number  of  seconds  required  for  the  body  to  fall 


250  ELEMENTARY  ALGEBRA 

192  yards,  or  576  feet,  substitute  576  for  s  in  (4)  ;  then, 

576  =  16  fi.  (6) 

Solving  (6)  for  t,  t  =  Q.  (7) 

Hence,  6  seconds  is  the  time  required  for  the  body  to  fall  192 

yards. 

EXERCISE   89 

Write  each  of  the  statements  in  the  first  five  examples 
in  the  form  of  an  equation. 

1.  The  area  ^  of  a  circle  varies  as  the  square  of  its 
radius  B. 

2.  The  volume  F  of  a  sphere  varies  as  the  cube  of  its 
diameter  D. 

3.  The  number  iV  of  articles  which  can  be  bought  for 
a  fixed  sum  aS'  varies  inversely  as  the  cost  P  of  one  article. 

4.  The  weight  w  of  a  body  varies  as  its  mass  m. 

5.  The  speed  v  of  a  falling  body  starting  from  rest 
varies  as  the  time  t  during  which  it  falls. 

6.  If  i/  varies  as  x,  and  ^=20  when  x  =  5,  find  the 
value  of  «/  when  a:  =  10. 

7.  If  1/  varies  inversely  as  x,  and  «/  =  20  when  x  =  5, 
find  the  value  of  i/  when  a;  =  100. 

8.  If  p  varies  as  q,  and  p  =  —  2  when  ^  =  —  3,  find  the 
equation  connecting  p  and  q. 

9.  If  m  varies  inversely  as  n,  and  m  =  30  when  n  =  6, 
find  m  when  n=9. 

10.  If  7  bu.  of  corn  are  worth  14.90,  what  are  3  bu.  of 
corn  of  the  same  quality  worth? 

Suggestion.     Employ  direct  variation. 

11.  If  10  men  can  do  a  piece  of  work  in  15  days,  how 
long  will  it  take  13  men  to  do  the  same  work? 

Suggestion.     Use  inverse  variation. 


CHAPTER   IX 


GRAPHS 

186.  Graphical  representation  of  the  relation  between 
two  variables.  It  is  convenient  to  represent  correspond- 
ing values  of  two  related  variables  by  points  in  a  plane. 
Such  a  representation  is  said  to  be  graphical.  When  a 
sufficient  number  of  points,  which  represent  pairs  of  cor- 
responding values  of  two  related  variables,  have  been 
marked  in  the  plane,  a  glance  at  the  diagram  will  reveal 
to  the  eye  the  corresponding  changes  in  the  two  variable 
quantities. 

For  the  purpose  of  graphical  work,  paper  ruled  in  small 
squares,  as  in  the  ac- 
companying diagram,  is 
used.  Two  of  the  ruled 
lines  of  the  paper,  one 
horizontal  and  the  other 
vertical,  are  selected  as 
axes  of  reference.  The 
point  0,  at  which  the 
axes  cross  each  other  is 
called  the  origin.  The 
axes  of  reference  are 
called  axes  of  coordi- 
nates. The  horizontal 
axis  X^X  is  called  the  jc-axis,  and  the  vertical  axis  YY'  is 
called  the  y-sxis. 

251  • 


— 

— 

— 

— 

Y 

^ 

"F 

/ 

n 

X 

1 

J 

_ 

Y 

252 


ELEMENTARY  ALGEBRA 


Any  point  in  the  plane  has  two  coordinates;  namely, 
its  abscissa,  represented  by  x^  and  its  ordinate,  repre- 
sented by  y.  The  abscissa  of  a  point  is  its  horizontal 
distance  from  the  ^-axis.  The  ordinate  of  a  point  is 
its  vertical  distance  from  the  a;-axis.  All  horizontal  dis- 
tances measured  to  the  right  from  the  y-axis  are  regarded 
as  positive,  and  consequently  (see  section  31),  all  hori- 
zontal distances  measured  in  the  opposite  direction  from 
the  y-axis  are  negative.  All  vertical  distances  measured 
upwards  from  the  x-axis  are  regarded  as  positive,  and  con- 
sequently, those  measured  in  the  opposite  direction  from  the 
X-axis  are  negative. 

187.  A  point  fixed  by  its  coordinates.  The  position  of 
a  point  in  a  plane  is  fixed  by  its  coordinates. 

Thus,  the  position  of  the  point  P  whose  coordinates  are  a:  =  2 
and  ^  =  —  1  is  fixed.  The  sign  of  its  abscissa  shows  that  the  point 
P  is  to  the  right  of  the  ?/-axis  and  the  sign  of  its  ordinate  shows  that 


Y 

P                        0                             X 
_,_ 

Y' 


it  is  below  the  ar-axis.  To  actually  find  the  position  of  P,  we  count 
on  the  ar-axis  2  squares  to  the  right  of  the  origin,  since  the  value 
of  the  abscissa  of  P  is  2 ;  then  we  count  downward  from  the  a;-axis 
1  square,  since  the  value  of  the  ordinate  of  P  is  —  1. 


GRAPHS 


253 


188.  Notation.  The  coordinates  of  a  point  are  desig- 
nated by  the  symbol  (x^  y). 

Thus,  the  point  (2,  —  3)  means  the  point  whose  x  (abscissa)  is  2 
and  whose  y  (ordinate)  is  —  3.  It  should  be  observed  that  in  such  a 
symbol  as  (2,  —  3)  the  number  written  first  is  always  the  abscissa. 

189.  Plotting  a  point.  The  marking  of  the  position  of 
a  point  on  the  diagram  is  called  plotting  the  point. 

ILLUSTRATIVE  EXAMPLES 

1.    Plot  the  point  (-2,4). 

Solution.  For  convenience,  we  take  five  divisions  on  the  coor- 
dinate paper  as  the  unit  of  measure.     The  abscissa  being  —  2,  we 


V 

n 

^j 

D 

• 

0 

X 

\ 

n/ 

. 

count  10  squares  (2  units)  to  the  left  of  the  origin  along  the  a:-axis  to 
the  point  B ;  then,  since  the  ordinate  is  +  4,  we  count  20  squares 
(4  units)  upwards  from  the  x-axis,  locating  the  position  P  of  the  point 
(—2,  4).     We  mark  the  position  of  P  by  a  small  ring. 


254 


ELEMENTARY  ALGEBRA 


2.    Plot  the  point  (1.25,  .5). 

Solution.     For  convenience,  we  take  20  divisions  on  the  coordi- 
nate paper  as  the  unit  of  measure.    Since  both  coordinates  are  positive, 


Y 

" 

- 

n 

1 

- 

"■ 

~ 

L 

(^a| 

A 

1/ 

0 

X 

/ 

1 

- 

- 

we  count  to  the  right  along  the  a:-axis  and  upwards  from  the  ar-axis. 
Counting  25  squares  (1.25  units)  to  the  right  along  the  a:-axis  and  then 
10  squares  (.5  units)  upwards  from  the  a;-axis,  we  locate  the  position 
P  of  the  required  point. 


EXERCISE  90 

(If  convenient,  use  graph  paper  in  plotting  the  graphs  in  the 
examples  of  this  exercise.  If  such  paper  is  not  available,  draw  two 
axes.  Use  an  appropriate  unit  of  distance,  as  ^  inch,  -J  inch,  J^  inch, 
or  1  centimeter). 

1.  Plot  the  points  (4,  3);   (-4,3);   (4,-3);   (-4, 
-3). 

2.  Plot  the  points  (-2,  1);   (-2,  -1);    (4,   -4); 
(4,1). 

3.  Plot  the  points  (3,0);  (0,-3);   (0,3);  (-3,0). 


GRAPHS  255 

4.  Plot  the  points  vl(4, 3);  ^(-4,4);  C(7,0).  Con- 
nect the  points  A,  B^  C  with  straight  lines  forming  a  tri- 
angle whose  vertices  are  the  given  points. 

5.  Construct  the  quadrilateral  whose  vertices  taken 
in  order  are  the  points  ^(3, 4)  ;  5(  -  2,  6) ;  (7(  -  4,  —  1) ; 
i)(4,  -2). 

6.  On  what  straight  line  are  all  points  located  which 
have  for  their  abscissa  0? 

7.  On  what  straight  line  are  all  points  located  which 
have  for  their  ordinate  0? 

8.  What  are  the  coordinates  of  the  origin?  Plot  the 
point  (0,  0). 

9.  Draw  the  triangle  whose  vertices  are  (0,-1); 
(0,  +1);   (2,0). 

10.  Plot  the  points  (-3,  -9);  (-2,  _6);(-l,  -3); 
(1,  3);  (2,6);  (3,9).  Do  these  points  appear  to  the  eye 
to  be  scattered  at  random  over  the  diagram?  How  do 
they  appear  to  lie? 

190.  A  function.  In  many  of  the  problems  of  elemen- 
tary algebra  one  or  more  pairs  of  related  variables  occur 
[see  examples  1-5,  exercise  89,  also  section  183].  As 
another  illustration  of  two  related  variables,  let  A  repre- 
sent the  age  of  a  boy  and  W  his  weight.  In  general,  as 
A  changes  it  is  evident  that  W  also  changes ;  this  fact 
may  be  expressed  by  stating  that  the  weight  of  the  boy 
varies  with,  and  depends  on,  his  age. 

If  one  variable  varies  with  another  so  that  when  a  value  of 
07ie  18  given,  a  corresponding  value  of  the  other  is  determined, 
the  second  variable  is  called  a  function  of  the  first. 

Thus,  from  the  indeterminate  equation  y  +  2  x  =  5,  we  derive 
y  =  5  —  2  X.     Here  y  is  so  related  to  x  that  its  value  is  determined 


256 


ELEMENTARY  ALGEBRA 


for  any  given  value  oi  x;  y  is,  therefore,  a  function  of  x.  Also,  the 
area  of  a  circle  is  a  function  of  its  radius.  If  A  represents  the  area 
and  R  the  radius  of  a  circle  and  tt  the  well-known  constant  whose 
value  to  four  places  of  decimals  is  3.1416,  then  the  relation  between 
the  area  of  a  circle  and  the  radius  is  expressed  algebraically  by  the 
equation  A  =  ttR^ 

191.  Graph  of  a  function.  When  a  simple  relation  be- 
tween two  variables  is  given,  as,  for  example,  the  relation 
expressed  by  the  equation  i/  =  S  x,  numerical  values  may 
be  assigned  to  x  and  the  corresponding  values  of  y  found. 
On  plotting  the  points  which  represent  the  different  pairs 
of  corresponding  values  of  the  variables,  these  points  are 
found  to  lie  on  a  definite  straight  line  or  curve ;  this 
straight  line  or  curve  is  called  the  graph  of  the  function. 
It  is  also  convenient  to  speak  of  the  graph  of  an  equation, 
an  expression  which  means  that  the  coordinates  of  the 
points  plotted  and  through  which  the  line  or  curve  passes, 
satisfy  the  equation. 

ILLUSTRATIVE   EXAMPLE 

Draw  the  graph  of  3  a;  -h  5  for  values  of  x  between  x=  —4 
and  a:  =  +  3. 

Solution.     Denoting  the  function  by  y,  we  have 

y  =  3x-\-5  (1) 

We  now  make  a  table  showing  the  corresponding  values  of  x  and  y 
for  integral  values  of  x  between  a;  =  —  4  and  a:  =  +  3,  as  follows : 


X 

-  4 

-3 

-2 

-1 

0 

1 

2 

3 

3x 
5 

-12 
5 

-  9 
5 

-6 
5 

-3 
5 

0 
5 

3 
5 

6 
5 

9 
5 

y 

-  7 

-4 

-1 

+  2 

5 

8 

11 

14 

Plotting  the  points  which  represent  the  values  of  x  and  y  (see 
diagram),  these  points  appear  to  lie,  and  in  fact  do  lie,  on  the  straight 


GRAPHS 


257 


line  PQ.     This  straight  line  is  the  graph  of  the  function  3  x  +  5,  and 
is  also  the  graph  of  the  equation  3/  =  3  a:  -f  5. 

Any  values  of  x  and  y  which  satisfy  equation  (1)  are  the  coordi- 
nates of  a  point  on  the  graph  of  equation  (1).     Conversely,  we  as- 


^ 

— ' 

~" 

■— 

~" 

^ 

Y 

■" 

n 

/ 

V 

1 

/ 

1 

/ 

1 

/ 

^ 

/ 

1 

/ 

1 

y 

0 

Y 

J 

/ 

1 

1 

/ 

T 

Y 

/ 

.J 

sume  that  the  coordinates  of  any  point  on  the  graph  of  the  equation 
will  satisfy  the  equation. 

Note.  The  expressions,  construct  the  graph  of,  obtain  the  graph 
of,  draw  the  graph  of  graph  the  function,  plot  the  curve,  mean  the 
same  thing. 

EXERCISE  91 

Draw  the  graphs  of  the  following  equations  for  values 
of  X  between  a:  =  —  4  and  a;  =  -f  4 : 

1.    1/  =  X.  2.    y  =  —  X.  3.    «/  —  2  a:  =  0. 

4.    y  +  4  a:  =  0.         5.    y  =  3  a;.  6.    y  =  a;  +  1. 


258  ELEMENTARY  ALGEBRA 

7.    y  =  x-\-1.         8.    y=-x^-1.         9.   y  =  ^x  +  1, 
10.    2x+Zy=5.  11.   3  2^—2^=1.       12.   5x  —  2t/  =  4, 

13.    2:+l/  +  l  =  0.   14.    ^+|-  3  =  0.  15.  3a:-|-4z/  +  7  =  0. 

192.  Graph  of  a  linear  equation  in  two  variables.     Any 

simple  equation  in  two  unknowns,  as  x  and  3/,  can  be  re- 
duced to  the  form  ax-^  by  -\-  c=  0,  The  graph  of  such  an 
equation  is  always  a  straight  line.  It  is  for  this  reason 
that  the  simple  equation  ax  -\-  by  -{-  c  =  0  is  called  a  linear 
equation. 

Note.  It  will  be  assumed  here  that  the  graph  of  any  simple  equa- 
tion in  two  unknowns  is  a  straight  line.  The  proof  of  this  fact  is 
given  in  analytical  geometry. 

193.  Graphs  of  simultaneous  linear  equations  in  two  un- 
known numbers.  When  the  straight  lines  which  are  the 
graphs  of  two  linear  equations  in  two  unknown  numbers 
are  plotted  on  the  same  diagram,  they  will,  if  the  given 
equations  are  independent  and  consistent,  intersect  in  one 
point.  The  coordinates  of  the  point  of  intersection  satisfy 
both  equations  and, .  therefore,  together  constitute  the 
solution  of  the  given  system. 

ILLUSTRATIVE  EXAMPLE 

Construct  the  graphs  of  the  equations  2y  —  a?-|-4  =  0 
and  2  i/ -I- 3  a;  — 4  =  0.  Estimate  from  the  diagram  the 
coordinates  of  their  point  of  intersection.  Find  whether 
or  not  these  estimated  coordinates  satisfy  the  equation. 

Solution. 

2y-a;  +  4  =  0.  (1) 

2y  +  3x-i  =  0.  (2) 

Solving  (1)  for  y,  2^=1-2-  (8) 

Solving  (2)  for  y,  y  =  =^  +  2.  (4) 


GRAPHS 


259 


Assigning  to  x  in  (3)  any  two  values,  as  a:  =  0,  and  x  =  4,  we  have 

from  (3),  when:.  =  0,     2,  =  -  2,  (5) 

when  a;  =  4,     y  =  0.  (6) 

Similarly,  assigning  to  x  in  (4)  any  two  values,  as  x  =  0,  and  x  =  1, 

we  have  from  (4),  u  n  o  x^x 

^  ^'  when  X  =  0,     y  =  2,  (7) 

when  X  =  \,     y  =  \-  (8) 

Since  we  know  that  the  graph  of  (3)  is  a  straight  line  [§  192, 
note],  it  is  necessary  to  plot  two  points  only  and  then  draw  the 
straight  line  through  these  points. 


Y 

\ 

\ 

■ 

^ 

\ 

V 

^ 

y^ 

X' 

0 

\ 

^ 

^ 

X 

^ 

^ 

\ 

^ 

\ 

\ 

^ 

Y' 

\ 

\ 

\ 

Plotting  the  points  (0,  -  2)  and  (4,  0),  the  graph  of  (3)  is  ob- 
tained by  drawing  the  straight  line  through  these  points.  In  like 
manner,  plotting  the  points  (0,  2)  and  (1,  |)  the  graph  of  (4)  is  the 
straight  line  joining  these  points. 

By  inspection,  the  graphs  of  the  two  equations  appear  to  intersect 
in  the  point  P  {x  =  2,  y  =  —  1).  Substituting  these  values  of  x  and  y 
in    (1)  and  (2),  we  have 

2(  -  1)  -  2  +  4  =  0.  (9) 

2(-l)+6-4  =  0.  (10) 

Hence  the  solution  of  (1)  and  (2)  is  x  =  2,  y  =  —  1. 


260  ELEMENTARY  ALGEBRA 


EXERCISE 


Construct  the  graphs  of  the  equations  in  each  of  the 
following  systems.  For  each  system  determine  by  in- 
spection approximate  values  of  the  unknown  numbers 
which  will  satisfy  the  equations. 

"••    \x-7/  =  ^.  \        y  =  2, 

/a;-h2i/=3,  r22:  +  3y  =  3, 

\^x-1y==l,  l3a:  +  42/  =  4. 

■3a;-hy  =  2,  1^  +  32^  =  6, 

2/=3.  •    \2:  +  2^  =  3. 


3. 


r3a;  + 
I2:r- 


194.  Graphs  of  dependent  linear  equations  in  two  un- 
knowns. Two  linear  equations  in  two  unknowns  which 
have  more  than  one  solution  in  common  must  have  all 
solutions  in  common,  and,  therefore,  they  are  dependent 
equations.  This  follows  directly  from  the  fact  that  the 
graph  of  any  such  equation  being  a  straight  line,  it  is  com- 
pletely determined  when  any  two  points  on  it  are  known. 

Thus,  the  equations  x  ■{■  y  =  2  and  3  a:  +  3  y  =  6  have  in  common 
the  solutions  x  =  0,  y  =  2  and  x  =  2,  y  =  0,  and  the  graph  of  each  is 
a  straight  line  through  these  two  points.  The  coordinates  of  every 
point  on  this  line  satisfy  both  equations. 

195.  Inconsistent  equations.  The  graphs  of  two  incon- 
sistent linear  equations  in  two  unknowns  furnish, a  simple 
geometrical  reason  for  the  existence  of  such  systems  of 
equations. 

Thus,  the  graphs  of  the  inconsistent  equations 

x  +  y  =  2  (1) 

x  +  y  =  l  (2) 

are  found  to  be  parallel  straight  lines. 

In  the  accompanying  diagram  two  divisions  on  the  coordinate  paper 
are  taken  as  the  unit  of  measure.     The  graph  of  (1)  is  the  straight 


Ren6  Descartes  (1596-1650)  was  the  first  of  the  modern  school 
of  mathematics.  He  was  educated  at  the  Jesuit  School  of  LaFleche, 
France.  Descartes  is  famous  as  a  mathematician  and  as  a  phi- 
losopher. The  introduction  of  the  fundamental  ideas  underlying  the 
graphical  representation  of  related  variables  was  his  greatest  con- 
tribution to  mathematical  science.  The  introduction  of  indices  as 
now  used   was  due  to  Descartes. 


GRAPHS 


261 


line  which  passes  through  the  points  (0, 2)  and  (2,  0),  and  the  graph 
of  (2)  is  the  straight  line  which  passes  through  (0,  1)  and  (1,  0). 
These  lines  are  evidently  parallel. 


Y 

\ 

\ 

\ 

\ 

^J 

\ 

\ 

\ 

\ 

^ 

\ 

\ 

X' 

O 

\ 

\ 

X 

\ 

\ 

\ 

\ 

Y' 

\ 

\ 

\ 

\ 

In  order  that  two  linear  equations  in  two  unknowns 
may  have  a  solution  in  common,  it  is  necessary  that  their 
graphs  should  have  a  point  in  common.  Two  parallel 
lines  have  no  common  point  and,  therefore,  the  equations 
of  which  the  parallel  lines  are  the  graphs  have  no  solution 
in  common;  they  are,  therefore,  inconsistent  equations. 

EXERCISE  93 

Construct  the  graphs  of  the  following  systems  of 
equations. 

a;  -  2  =  0, 

z/  -  3  =  0. 


2. 


3. 


x=^2y, 

y  -  1  =  0. 
x-\-  y  =  l, 
2  2^  +  2  ^/  =  2. 


CHAPTER   X 
POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS 

196.  Rational  number.  The  rational  operations  in 
algebra  are  addition,  subtraction,  multiplication,  and  divi- 
sion [section  79].  Any  number  which  is  either  a  positive 
integer  or  can  be  obtained  from  the  positive  integers 
by  the  rational  operations  is  called  a  rational  number. 
Hence,  a  rational  number,  when  expressed  in  its  simplest 
form,  is  either  a  positive  or  negative  integer  or  a  positive 
or  negative  fraction  with  integral  numerator  and  integral 
denominator. 

Thus,  2,    3.16,  —  7,  and  —  |  are  rational  numbers. 

Also,  +  y/i  is  a  rational  number,  since  when  expressed  in  its 
simplest  form,  it  has  a  rational  value,  namely,  2.  However,  finding 
the  square  root  of  a  number  is  not  one  of  the  rational  operations  of 
algebra. 

197.  Irrational  number.  It  is  necessary  to  add  to  the 
system  consisting  of  all  rational  numbers,  certain  other 
numbers  which  are  not  rational.  For  instance,  in  men- 
suration it  is  necessary  to  say  that  the  length  of  the 
diagonal  of  the  square  is  equal  to  the  length  of  its  side 
multiplied  by  the  V2;  yet  there  is  no  rational  number 
whose  square  is  equal  to  2  ;  that  is,  V2  is  not  a  rational 
number.  We  shall  assume  that  there  is  a  definite  positive 
number  whose  square  is  equal  to  2.  We  call  such  a  num- 
ber an  irrational  number. 

Thus,  V3,    1  —  V2,  — - ,  V2  +  V3    are  irrational  numbers. 

V2 

262 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     263 

Note.  When  we  say  that  we  assume  that  there  is  a  definite  posi- 
tive number  whose  square  is  equal  to  2,  we  are  extending  the  meaning 
of  the  word  number.  The  word  number,  heretofore,  has  meant  rational 
number.  Now  when  we  use  the  word  number  we  shall  mean  irra- 
tional number  as  well  as  rational. 

Remark.  We  do  not  call  such  an  expression  as  V—  2  an  irra- 
tional number.  This  expression  is,  for  the  present,  without  meaning. 
The  square  of  either  a  positive  or  a  negative  number  is  necessarily 
positive  ;  hence,  there  is  no  positive  or  negative  number  whose  square 
is  -2. 

Rational  numbers  and  irrational  numbers  are  necessarily  positive 
or  negative. 

198.   Fundamental    identities   involving  powers.      The 

exponents  used  in  the  identities  of  this  section  are  posi- 
tive integers. 

From  section  58  we  have, 

(T    a"  =  tf"+".  (I) 

By  definition,  section  14, 
(a6)"'  =  ab  ■  ab  •  ab  •  ■■•  to  m  factors 

=  a  ■  a  a  -  •••  to  m  factors  x  b  •  b    b  -  .••  tow  factors 

That  is,  {abY  =  cTlf.  (II) 

Remark.  Since  {abc)'^  =[(a6)c]'"  =(a6)'"c'*  =  a"'6'"c'",  it  follows 
that  an  identity  similar  to  (II)  holds  for  a  power  of  the  product  of 
any  number  of  factors. 

By  definition,section  14,  (cry  =  a"'-  a*"  •  a*"  . ...  to  w factors 

ffn-\-m-\-rn-\ to  fl terms 

=  a'"". 
In  like  manner,  (a")*"  =  a*"". 

That  is,  {ary  =  (a")'"  =  o^".  (Ill) 

From  identity  (II),  (a'"i»*)p  =  («»") p(5") p. 

From  identity  (III),    (a"*)p(5~)p=  a'^pft^p. 
Therefore,  {oTlfy  =  aTflfP.  (IV) 


264  ELEMENTARY  ALGEBRA 

EXERCISE  94 

(Solve  as  many  as  possible  at  sight.) 

Find  the  results  of  the  indicated  operations  by  using 
the  identities  of  section  198. 

1.    23.22.  2.    28.22.2.  3.    (2.3)2. 

4.     (2.3.5)8.  5.     (22)3.  6.     (33)2. 

7.     (23)3.  8.      [(-2)2]3.  9.      [(_2)3]2. 

10.      [(-2)3]8.  11.      (22.33)2.  12.      (2.32.52)2. 

13.    ai5  .  a}^.  14.    a^  -a^  •  a*.  15.    (aby. 

16.    (2a)3.  17.    (-3a)3.  18.    (2aby. 

19.    (-3a6)2.  20.    (abcdy.  21.    (a^^y, 

22.     (2253)3.  23.      [(-2)8a253]2. 

24.     (-3a253^2^4g6)4.  25.      [(«  +  *)2]3. 

26.    (a  +  by(a-{-b).  27.    (a-by(a-by. 

28.  (a-by(a-{-by. 

Suggestion,     (a  -  hy{a  +  hy  =[(«-ft)(a  +  &)p. 

29.  (a  -  l)2(a -f  1)2.  30.     [2(a  H- 6)2((?  -  (^)8]2. 
31.     [2a5(c+(^)2(e-/)3]3.  32.      [(-3)2]3.(33)2. 

33.  Show  that  (  —  J  =  — ,  and,  in  general,  that  (- j  =- — 

\bj       b^  \bj       6*" 

Suggestion.  (-Y  =  ?  .  ^  .  ?  ...  to  m  factors.    ^  [§  139] 

34.  Show  that  (f  )8(  j)8  =  (I  X  |)8,  and,  in  general,  that 

W  W       \bd)  ' 
Suggestion.  {^-\{'-Y^  ««  ^  ^  ««^  ^  ^^ 

35.  (1)2(1)2(1)2.  36.     (|)««(f)^(|)«^. 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     265 

199.  Roots.  When  a  number  is  the  product  of  five 
equal  factors,  one  of  these  factors  is  called  a  fifth  root  of 
the  number.  In  general,  when  a  number  is  the  product  of 
n  equal  factors,  one  of  those  factors  is  called  an  nth  root 
of  the  number. 

Thus,  if  m^  =  a,  then  m  is  a  fifth  root  of  a,  which  is  expressed  by 
m  =  Va. 

The  two  equations, 

m^  =  a  (1)    and   m  =  v'a  (2) 
express,  therefore,  the  same  relation  between  a  and  m.     By  substitut- 
ing the  value  of  m  from  equation  (2)  in  equation  (1),  we  have  the 
identity  (Va)^  =  a.     In  general,  when  n  is  any  positive  integer  we 
have,  by  definition. 

Note.     The  expression  Va  is  read  the  nth  root  of  a. 

200.  Radical.  An  indicated  root  of  any  number  is 
called  a  radical  expression,  or  simply  a  radical. 

Thus,  V3,   </27,  Vr»  v^«T^  are  radicals. 

201.  Index  of  a  root.  The  number  indicating  the  root 
to  be  taken  is  called  the  index  of  the  root.  The  index  of 
a  radical  is  always  an  integer. 

202.  Radicand.  The  number  or  expression  under  the 
radical  sign  is  called  the  radicand. 

203.  Like  roots  and  unlike  roots.  Two  roots  are  said 
to  be  like  or  unlike  according  as  their  indices  are  equal 
or  unequal. 

Thus,  y/a  and  Vb  are  like  roots ;  Va  and  y/a  are  unlike  roots. 

204.  Principal  root.  Any  positive  number  a  has  one 
and  only  one  positive  nth  root.  This  root  is  called  the 
principal  nth  root  of  a. 

Thus,  the  principal  square  root  of  4  is  2,  the  principal  cube  root 
of  27  is  3,  and  the  principal  square  root  of  2  is  +  V2. 


266  ELEMENTARY  ALGEBRA 

When  the  index  of  a  radical  is  an  odd  number  and  the 
radicand  is  negative,  there  is  one  and  only  one  negative 
root  and  no  positive  root.  This  negative  root  is  called  the 
principal  nth  root  of  the  negative  number  in  the  radicand. 

Thus,  V  —  8  can  have  no  positive  value,  since  the  cube  of  a  positive 
number  is  positive.  It  has,  however,  one  negative  value,  namely  —  2, 
since  an  odd  power  of  a  negative  number  is  negative.  Also,  the 
principal  root  of  \/-  27  is  -  3,  that  of  y/ -  82  is  -  2,  that  of  \/-  2 
is  —  V2 ;  and  if  n  be  any  odd  integer  that  of  \/—  a^  is  —  a. 

205.  A  property  of  positive  numbers.  Two  positive 
7iumhers  are  equal  if  any  like  powers  of  these  numbers  are 
equal. 

For,  a  positive  number  has  one  and  only  one  positive 
root,  namely,  its  principal  root. 

Thus,  if  a  is  positive  and  a^  =  3^,  then  a  is  equal  to  3,  since  a  is  the 
principal  cube  root  of  a^,  or  of  27. 

206.  Notation.  In  what  follows  in  this  chapter  the 
letters  a,  5,  c,  and  so  on,  will  represent  positive  numbers 
or  literal  expressions  which  have  positive  values,  except 
when  otherwise  stated.  By  the  root  of  a  number  we  shall 
mean  its  principal  root ;  that  is,  its  one  positive  root  when 
the  radicand  is  positive  and  its  one  negative  root  when 
the  radicand  is  negative  and  the  index  an  odd  integer. 
[See  section  204.] 

Thus,  'v^  =  +  3,  and  \/^^27  =  -  3. 

207.  Surds.  A  surd  is  an  irrational  number  which  is  a 
root  of  a  rational  number. 

Thus,  V5  and  V3  are  surds ;  V4  and  v^27  are  radical  expressions, 
but  they  are  not  surds. 

208.  Order  of  a  surd.  The  order  of  a  surd  is  the  index 
of  the  root  involved  in  the  expression. 

Thus,  V2  is  a  surd  of  the  second  order,  or  a  quadratic  surd,  y/o  is  a 
surd  of  the  third  order,  or  a  cubic  surd. 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     267 

209.  Fundamental  identities  involving  roots.  Radicals 
are  transformed  and  combined  according  to  certain  rules. 
These  rules  may  be  expressed  in  the  form  of  algebraic 
identities. 

(%)"  =  a  (I) 

Identity  (I)  is  simply  the  definition  of  a  root  as  ex- 
pressed in  algebraic  symbols  [section  199]. 

EXERCISE  95 

State  at  sight  the  value  of  each  of  the  following: 
1.    (V2)2.  2.    (V3)2. 

3.    (V^)2.  4.    (^2)3. 

5.    (-^31)3.  6.    (</5y. 

7.    (^310)6.  8.    (^/^^)7. 

9.  (-s/ly.  10.  (</2iy. 


11.    (-y/a+by.  12.    (Va2_62)3. 

13.  (^V^a-^byy,  14.  (^/(a  +  byy. 

15.   VT  .  VT.  16.   Vi  •  Vi. 

17.     V^  X  ^^  X  ^"=7.         18.     Vx-^/x-  </i. 

^^='V^P.  (II) 

From  identity  (II)  we  may  infer  that : 

The  value  of  a  radical  is  not  changed  if  its  index  and  the 
exponent  of  its  radicand  are  both  multiplied  by  the  same  posi- 
tive integer. 

Conversely,  we  may  write  identity  (II)  thus: 

From  the  second  form  of  identity  (II)  we  have  : 

The  value  of  a  radical  is  not  changed  if  its  index  and  the 

exponent  of  its  radicand  are  both  divided  by  the  same  positive 

integer  which  is  a  factor  of  each. 


268  ELEMENTARY  ALGEBRA 

Note.     An  important  particular  case  of  §  209,  I,  is  expressed  by 
the  identity, 


Thus,  Va^P  =  ^(apy  =  op. 

Similarly,  va*  =  a^,  and  Va^  =  a*. 

ILLUSTRATIVE  EXAMPLES 

1.  Vo^  =  Va^  and  conversely,  V«^  =  Va^. 

2.  Va^  =  a,  and  conversely,  a  =  Va^. 

3.  ^/a3^=  a"*,  and  conversely,  a"»  =  ^/c^. 

The  proof  of  identity  (II)  is  as  follows ; 

By  identity  (I),         (Va^)'*^  =  0*"^. 
Also,  by  identity  (III)  of  §  198, 

=  (a-)p  [§209  (I)] 

=  a^p.  [§198(111)] 

We  have  now  shown  that  the  (w^)th  power  of  the  ex- 
pression in  the  first  member  of  identity  (II)  is  equal  to 
the  same  power  of  the  expression  in  the  second  member. 
It  follows  by  the  principle  of  section  205  that  the  two 
expressions  are  equal. 

EXERCISE  96 

(Solve  as  many  as  possible  at  sight.) 
Simplify  the  following:        • 

1.     ^/A  2.    -v^.  3.    ^'^. 

4.  ^(2)8.  '  5.  </(a  +  by.    6.  ^/(c-j-dy^, 

7.    ^3^  8.    ^/^^r^^.  9.    ^^T^. 

10.    V-(a  +  6)6.  11.   .J/^.  12.    </(a  +  by. 

13.    \/(a  -  6)16.  14.    ^^.  15.    </a^. 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     269 

Vab=\^a^i.  (Ill) 

From  identity  (III)  we  infer  that : 

An^  root  of  the  product  of  two  or  more  factors  is  equal  to 
the  product  of  the  like  roots  of  the  factors. 

Note.  In  identity  (III)  there  are  two  factors  and  in  the  statement 
of  the  principle  that  follows  there  are  tioo  or  more  factors.  Here, 
as  in  all  similar  cases,  what  is  true  of  the  product  of  two  factors  is 
true  in  general.  This  follows  directly  from  the  fact  that  by  grouping 
factors  any  product  may  be  expressed  in  the  form  (a&). 

Thus,  in  the  case  of  four  factors, 


VaM=  y/{ah){cd) 
=  VabV^ 
=  (Va  Vb)  {V~c  Vd) 
—  yja  Vh  Vc  yjd. 

Writing  the  converse  of  identity  (III),  we  have, 

This  second  form  of  identity  (III)  may  be  stated  in 
words  as  follows: 

The  product  of  like  roots  of  two  or  more  numbers  is  equal 
to  the  like  root  of  their  product. 

ILLUSTRATIVE   EXAMPLES 


1.    V8  =  V4  x2  =  V4  V2  =  2V2. 


2. 


^/54  =  V  27  X  2  =  a/33  ^2  =  3^/2. 


3.  V3a2=  VaW3  =  aV3: 

4.  ^4^2  =  ^=  ^P  =  2. 

5.  VaVa^Va^  =  -^a  x  a^  x  a^  =  \^a^  =  a. 

In  order  to  prove  identity  (III)  we  show  that  the  nth 
power  of  the  positive  number  expression  in  the  first  mem- 
ber is  equal  to  the  same  power  of   that  in  the  second. 


270  ELEMENTARY  ALGEBRA 

Thus,  _ 

By  identity  (I)  (y/ahy  =  ah. 

Also,  by  identity  (II),  section  198, 

{VaVby  =  {y/aY{y/hy,  and  by  section  209,  (1), 
=  ah. 
Since  (v^aft)"  and  {Vay/hY  are  each  equal  to  ah,  it  follows  that 
y/ab  =  V~^Vh. 

EXERCISE  97 

(Solve  as  many  as  possible  at  sight.) 
Simplify  : 

1.    VI2.  2.  V27.  3.  V32. 

4.    V128.  5.  V2^.  6.  V3^4. 

7.  V4a3.      Suggestion.  Vi^s  ^  V(4  a2)a. 

8.  V4^.  9.  V8^3.  10.  VT28"^. 
11.  V^.  12.  V47m2.  13.  Va25M#. 
14.    V72^.               15.  V80  ^254.            16.  V99^». 

17.  V127008.      Suggestion.     127008  =  25.34. 72. 

18.  V84672.  19.  V27  a6V. 
20.  V81  X  5  X  «355.  21.  V75%2. 
22.    V176 «2j2^4#.          .           23.    V2(^l^y. 

24.     V50(a+5)2(c+(^)4.  25.     Vl08a2  (5+^)3. 

26.     ^/T6.  27.     ^54. 

28.    ^2^128.  29.    \/625. 

30.     ^81.  31.     ^S/STS. 

32.  ^128l3. 

33.  \/— 16.      Suggestion.     \/^^^T6  =  -  \/2(2)8. 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     271 


34.    V  -  54.  35.    V  -  216  2^,         36.    V9  a^y^z^. 


37.  V-125a356.       38.  ^/U.  39.    V64a:5^io. 

40.  ^^^.      Suggestion,     v^^^  =  -  v/2  [§  204]. 

41.  -s/"^^.                     42.  x/"^.  43.     ^/- 32  2^3^. 
44.  ^^.                    45.  ^-X^^yz^.  46.     ^-2y. 

47.  ^-2:10.      Suggestion,     ^^^^^o  =  v^(- l)6a;io. 

48.  ^32^6.  49.     I^^4. 

50.  v/?2.  51.     ^214  •  «!*. 

52.  v'- 243^667.  53.    ^(>  +  i)^(<?  +  dy^. 


54.    V28  .  39  .  aV^io.  55.    </-(^x  +  yy{x-yy. 

i^ar^Va^.  (IV) 

From  identity  (IV)  we-  infer  that : 

Any  power  of  a  radical  is  obtained  hy  rkultiplying  the 
exponent  of  the  radieand  hy  the  exponent  of  the  required 
power. 

Conversely,  since  Va  is  one  of  the  m  equal  factors  of 
(Va)*^,  or  of  its  equal  -v/o^,  it  follows  that  \/a  is  the  Twth 
root  of  -v/a^.     Hence  : 

Rule.  To  find  the  root  of  the  radical^  the  exponent  of 
whose  radieand  is  exactly  divisible  by  the  index  of  the  root,, 
divide  the  exponent  of  the  radieand  by  the  index  of  the  re- 
quired root. 

ILLUSTRATIVE  EXAMPLES 

1.  (</8)3  =  (</23)3  =  </29  =  ^28~x^  =  </28  </2  =  22^2 
=  4</2. 

2.  (\^)6  =  ^=  42  =  16. 

3.  \/^=V^  =  a. 


272  ELEMENTARY  ALGEBRA 

The  proof  of  identity  (IV)  is  as  follows  : 
By  identity  (I),  section  209,  (v^a«)»=  a« 
Also,  by  identity  (III),  section  198,  [(v^a)*]"  =[(\/a)»]'«  =  a'". 
Since  (\/a*»)«  and  [(%)'«]«  are  each  equal  to  a"*,  it  follows  that 
(%)"»  =  v^^[§205]. 

EXERCISE  98 

In  simplifying  the  following,  use  as  many  of  the  pre- 
ceding identities  of  this  chapter  as  may  be  necessary. 

1.  (</49)2.  2.  (V2)4. 

3.  (V3)6.  4.  (Va)4. 

5.  (V3^3)8.  6.  (V2^)4. 

7.  (V3^)6.  8.  (V3^)5. 

9.  (V^ay,  10.  (\/^^2^)io. 

11.  (-^9)4.  12.  [\/2(a+5)]2. 

13.      [■^-3((?  +  C^)2]2.  14.     (^/2^)16. 

15.     (_^3TS)15.  16.     (-^'3T^2)6. 

17.    \/Vi.  18.     V'V9T2. 


19 


.   v^V—S.  20.    \/-</^32. 


(V) 


From  identity  (V)  we  infer  that : 

Ani/  root  of  a  fraction  is  equal  to  the  like  root  of  the  nu- 
merator divided  by  the  like  root  of  the  denominator. 
Conversely,  we  may  write  identity  (V)  thus  : 

V6     ^fr' 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     273 

From  this  second  form  of  identity  (V)  we  infer  that: 
Any  root  of  a  number  divided  by  the  like  root  of  a  second 
number  is  equal  to  the  like  root  of  the  fraction  whose  numera- 
tor is  the  first  number  and  whose  denominator  is  the  second. 

ILLUSTRATIVE   EXAMPLES 

3f^      ■^/'^      a2 


2. 


^6l2  .yp  b^ 

2^  =  ^/1  =  J5=:^=:^ 

•  V3     ^3     ^9      V9       3   * 

<^3  ^    /3  ^   3/12  ^  ^12  ^  </T2 

*  ^2     ^2      ^8        ^8         2    ' 

The  proof  of  identity  (V)  is  as  follows : 
By  identity  (I),  §  209,       (^^y  =  |. 
By  identity  in  example  33,  p.  264, 

Since  f -^-  J    and  (■;;—)     are  each  equal  to  -,  it  follows  that. 


274  ELEMENTARY  ALGEBRA 

EXERCISE  99 

Reduce  the  following  to  equivalent  fractions  having 
rational  denominators  [see  illustrative  examples  5  and  6, 
p.  273] : 


4. 


7. 


10. 


13. 


16. 


19. 


22. 


25. 


128. 


From  identity  (VI)  we  infer  that : 

Any  root  of  a  root  of  a  number  is  equal  to  that  root  of  the 
number  whose  index  is  the  product  of  the  given  indices. 
The  converse  of  identity  (VI)  shows  that  when  the  index 


V3 

V2 

2. 

vj. 

3. 

.V|. 

vj. 

5> 

vj. 

6. 

V7 
V8 

V3 

V5 

8. 

vTi 

V5 

9. 

VI- 

^a 

11. 

VS- 

12. 

VA- 

yJ^". 

14. 

</2 

15. 

^s- 

n. 

17. 

18. 

V?-  ■ 

20. 

21. 

\j6 

Va6 
Va 

23. 
26. 
29. 

■V2a? 
V3a 
'116  a* 

24. 
27. 
30. 

^86« 

sj-Sa^^ 
^  125^9 

5/32  aW» 
>'243c6 

/(a +  6)8^ 

^ie  +  dy 

V 

Va='"</a. 

(VI) 

POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     275 

of  a  given  root  is  a  composite  number,  it  is  possible  to  ex- 
press the  given  root  in  terms  of  simpler  roots. 

Thus,  \/625=VV625=:  \/25=5;  also,  v^  =  Vv^=  V^=aV5. 
ILLUSTRATIVE  EXAMPLES 

1.  -Wa=</a 

2.  -^(a4-^)2  =  ^^T^;  [§  209  (II)] 
or,          -^(a+by  =  \/V(a  +  5)2  =  -^^:^.          [§  209  (VI)] 

Remark.     Since  VVa="^a 

and  VV^^'^^a, 

it  follows  that  V^  =  V^. 

Thus,  V^2=v/V5  =  ^. 

Also,  V  V8  a6^»8c9  =  V  v^8^^6p^  =  V2  a%c^ 

=  y/a^V2bc  =  ac^/2bc. 

Remark.  The  proof  of  identity  VT  is  left  as  an  exercise  for  the 
student.     See  section  205. 

210.  Simplification  of  radicals.  A  surd  whose  radicand 
is  integral  is  said  to  be  in  its  simplest  form  when  the  index 
of  the  radical  is  as  small  as  possible  and  when  no  factor  of 
the  radicand  has  an  exponent  which  is  exactly  divisible  by 
the  index  of  the  root. 

Thus,  V2a  is  in  its  simplest  form :  v a^  is  not  in  its  simplest  form 
since  Va^  =  Va^  x  a  =  Va'^Va  —  « Va  ;  vTT^  is  not  in  its  simplest  form 
since  by  section  209  (II),  Va*  =  >Ja. 

From  identities  (HI)  and  (11),  note,  section  209,  we 
derive  the  following  rule  for  simplifying  surds  : 

Rule.  Make  the  index  of  the  radical  as  small  as  possible, 
then  if  the  exponent  of  aiiy  factor  of  the  radicand  is  divis- 
ible by  the  index,  divide  the  exponent  of  that  factor  by  the 
index  and  remove  the  factor  from  under  the  radical  sign. 

Thus,         Vbia^-^c^  =  y/2x  ^^a%^c^a%  =  3  a%^c </2~^, 


276  ELEMENTARY  ALGEBRA 

Conversely,  any  factor  of  the  coefficient  of  the  radical 
may  be  brought  under  the  radical  sign  and  be  made  a 
factor  of  the  radicand  provided  that  its  exponent  be 
multiplied  by  the  index  of  the  radical. 

Thus,  2  Vd^  =  y/WxSTi  =  Vl2^. 

Any  surd  whose  denominator  is  irrational  is  not  in  the 
simplest  form  for  the  approximate  numerical  calculation 
of  its  value.  This  may  be  shown  by  a  particular  example, 
as  follows :  , 

Find  an  approximate  value  of  '^-. 
Solution  (1)  ^|  =  V5  =  ^  =  ?f?  =  .816^ 

Solution  (2)  A(|  =  ^  =  l^=•816^ 

Solution  (3)  ^l  =  ^l  =  ±  =  ^^.MeK 

The  actual  work  required  in  solutions  (2)  and  (3)  is  greater  than 
that  required  in  solution  (1).  The  most  convenient  method  of  cal- 
culation, therefore,  is  that  which  proceeds  by  first  making  the  denom- 
inator of  the  fraction  rational. 

211.  Simplest  form  of  a  radical  expression.  A  radical 
expression  is  said  to  be  in  its  simplest  form  when  its  de- 
nominator is  rational  and  all  integral  radicands  which  it 
contains  are  reduced  to  their  simplest  forms. 

212.  Simplification  of  a  simple  fractional  surd.     A  surd 

whose  radicand  is  a  fraction  in  its  lowest  terms  is  simplified 
by  multiplying  the  numerator  and  the  denominator  of  the 
radicand  by  the  simplest  expression  which  will  make  its 
denominator  rational. 

Thus       ^^^^    ^  « /"TT  ^  i/3"^(5^  ^  v^lS^P  ^  \/l5^ 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     277 

213.  Similar,  or  like,  radicals.  Two  radicals  which, 
when  expressed  in  their  simplest  forms,  differ  only  in  their 
coefficients,  are  said  to  be  similar,  or  like. 


Thus,  V8a^  and  —  Vl8  ab^  are  similar,  since  their  simplest  forms, 
2  aV2ab  and  —  3  6\/2  ab,  differ  in  their  coefficients  only. 

EXERCISE   100 

1.  Show  that  V12,  V27,  v|,  VJ,  are  similar. 

2.  Arrange  the  following  radicals  in  sets  so  that  those 
in  the  same  set  shall  be  similar : 


^2ab^;   V4a353;   yj^-^;   ^9a%;   W9a%;  ^^-^. 

3.    Write  the  coefficient  as  a  factor  of  the  radicand  in 
each    of     the    following:     2V5;     3V2;    aV3;     2aV3; 

Suggestion.     2V5  =  V22x  5  =  V20.  [§210.] 


4.  Simplify  V(a2  —  b'^){a  +  6),  in  which  expression  a  is 
greater  than  b. 

5.  Which  of  the  following  radical  expressions  are  surds : 

V2?     ^/S?     VlH-V2?     \/Vl? 

Reduce  each  of  the  following  radicals  to  its  simplest 
form : 


6.    V72a62c3. 
9.    ^/W¥f. 


12 

^  y 


15.    y/^x^y^-^xh^ 


7.     </-82. 

10.    V^. 

8. 
11. 

14. 

-y 

^  X 

13.    \/32*yz«. 

^X 

Va:^  ^  2:2^, 

278  ELEMENTARY  ALGEBRA 

18.     VWMK 


17. 

^  27  aS 

19. 

Va2«. 

21. 

Va4«+i. 

23. 

Va2»+i. 

25. 

V?- 

37 

Jl?-Z  +  1 

^  ix+iy 

29. 

(<^)2.   . 

31. 

2 

20.    Va2«+i. 
22.    Va;3  +  2^3/. 
24.    Va2"+i2:. 


26. 


^9^. 


3 

V3* 
1 

^8' 


30. 


32. 


Given  V2  =  1.414,  V3  =  1.732,  V5  =  2.236,  calculate 
to  two  places  of  decimals  the  values  of  the  expressions  in 
examples  33,  34,  and  35. 

33.     ^'  34.     Vf  35.     V|. 

V3 

36.  Simplify  V7  x  VT  x  V6  x  V2  -s-  V3. 

37.  Simplify  V63  x  VT68 -i- V27. 

38.  Simplify  Vl4  x  VT5 -^  VM. 

39.  Simplify  V35  x  V6  -5-  V30. 

40.  Simplify  V12  x  V40  x  V20  x  V48  x  V24. 

41.  Simplify  (^/^)9x(VF)6x(</^)8. 

214.  Comparison  of  surds.  For  certain  purposes  it  is 
convenient  to  express  two  or  more  surds  in  terms  of  like 
roots,  as,  for  example,  when  finding  which  of  two  given 
surds  is  the  greater. 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     279 

Thus,  in  finding  which  of  the  two  surds  Vo  or  v^7  is  the  greater, 
we  make  use  of  identity  (II),  section  209,  to  transform  both  expressions 
into  surds  of  the  same  order  as  follows : 

It  is  now  evident  that  y/b  is  greater  than  V7. 

EXERCISE  101 

1.  Express  V5  and  ^10  as  surds  of  the  same  order. 

2.  Which  is  greater,  V5  or  VS? 

3.  Which  is  greater,  V3  or  V28? 

4.  Arrange  2\/27,  3\/4,  2V6  in  order  of  magnitude. 
Suggestion.     Place  the  coefficients  under  the  radical  signs. 

5.  When  two  or  more  surds  are  reduced  to  equivalent 
surds  having  a  common  index,  what  is  their  lowest  com- 
mon index? 

215.  Addition  and  subtraction  of  surds.  The  algebraic 
sum  of  two  or  more  surds  is  expressed  in  the  simplest 
form  when  each  surd  is  in  its  simplest  form  and  all  similar 
surds  are  combined  by  adding  their  coefficients. 

ILLUSTRATIVE  EXAMPLES 

1.  Add  5V8,  -4V50,  3V72,  -2V98,  and  VT28. 

Solution.     The  sum  of  5V8,  -  4V50,  3V72,  -  2V98,  and  Vl28 
=  10V2  -  20V2  +  18V2  -  14V2  +  8 VI 
=  (10-20  +  18-14  +  8)V2 
=  2V2. 

2.  Simplify     2Va5^- V9"^2_  yf^^^  gVPi^. 

Solution.     2  y/ab^  -  V9  ahy^  -  V4  a^xy  +  3  x/P^ 

=  2  xVab  —  3  yy^ah  —  2  ayJxy  +  3  h\/xy 
=  (2x-3y)y/ab-(2a-'db)y/xy. 


280 


ELEMENTARY  ALGEBRA 


Note.  The  sum  of  two  unlike  surds  cannot  be  expressed  as  a 
single  surd ;  such  a  sum  can  only  be  indicated. 

Thus,  the  expression  \/2  +  VS  is  in  its  simplest  form. 

Remark.  Particular  attention  should  be  called  to  the  fact  that 
V3  -f  V2  is  not  equal  to  V3  +  2,  or  VS. 


EXERCISE  102 

Simplify  each  of  the  following  expressions: 


1.  3V6-2V6  +  5V6.  2. 

3.  V50-fV32-VT8.    •         4. 

5.  3V60-V240-2V15.       6. 

7.  a/250  -  2^16  +  \/5¥.         8. 

9.  VT8-|V2+V^.  10. 

11.  V50  4-VT08+V98  +  3VT2. 


V5i-V24  +  Vl50. 

vT5-V27-2Vl2. 
V9^- VTSP  +  Va?. 

-i5,_V27+V|. 

V3  ^ 


12.   2V90-V176  + 


V160. 


13. 
14. 

15. 
17. 


V4M-  V9  ab^-{-  2  V4  ae^  -  V9  ab^  +  V4  aS-j-  V16  ahc\ 
\  V9^6  -  2  J^-  3x/'?  +  ^^^^. 

2^4  +  2-\/32-9^I.        16.    2^/4- 5^32 +  3\/108. 
VT8-A/24.  "  18.    ^6  +  ^1/4-^1. 


19.    V6W|-4Vy4-VJf.  20.    I^^  +  JV^I^. 
21.    H^l'^^^-I^l-I^A-  22.    a/T62^  +  2-v/32^. 
23.   x^l5f--V4S¥f-\-xi/V27. 
V'3^  -  2  V3^  +  V3^. 


24 


-•  >g-xa+%/i- 


26. 
27. 
28. 


■v/8  +  24  a  +  VsToTW. 


Vl2^^^^S¥-\-  V27  ftV  +  18  62 -  V48  (^x^-{-  32  c^. 
3  V3^  -  V48^  -  V3  a3  4-  6  a2  4-  3  a.  , 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     281 
29.    (a  -f  hy^ab  +  (a  -  hyVab  -  (a2  +  5^)  V^. 


30.    V2:3_2a:2_V4a:-8-V(a:  +  2)(a:2_4). 

216.  Multiplication  and  division.  The  rules  for  the 
multiplication  and  division  of  monomial  surd  expressions 
are  derived  from  identities  (III)  and  (V),  section  209; 
namely,  from  the  identities, 

VaV5  =  ^ah  and  -;r^  =  \t' 

Remark.  Observe  that  these  identities  contain  surds  of  the  same 
order  only.  Hence,  before  two  or  more  surds  are  combined  by  mul- 
tiplication or  division,  they  must,  when  necessary,  be  reduced  to  surds 
of  the  same  order.     See  section  214. 

ILLUSTRATIVE   EXAMPLES 

1.  5 V20  X  3 V45  =  10 V5  x  9  V5  =  90(  V5)2  =  450. 

2.  2V^x5v^^  =  2^^3x5-y^454^10^^^ 

=  lOah^/ab. 


3.    V6^V5=x/^  = 
6 


EXERCISE  103 

Simplify  each  of  the  following  expressions: 
1.    V5xVT0.  2.    VlO^VS. 

3.    axVa^  x  hy^l^.  4.    V6  X  V2. 

5.    V'6  ^  Vl2.    -  6.    axy/'c^  -j-  hyVa. 


282  ELEMENTARY  ALGEBRA 

7.  V5xV20.  8.  V90-hVT0. 

9.  Va  X  V6.  10.  2Va  X  3V6. 

11.  2^/3x3^/9.  12.  ^320-^\/5. 

13.  -v^24-^-v/3.  14.  2a/64-4^/8. 

15.,  ^18x-n/3.  16.  </|x^I 

17.  V3x\/2.  18.  V2xa/3. 

19.  a/4xV8.  20.  ^x^SPx^^^. 

21.  V^lix-s/S.  22.  Vi-V9. 

23.  Vf-f-Vf.  24.  1-^-5-1^. 

25.  20-5-V|.  26.  8V^-!-4^^. 

27.     ^m^2  ^  V8^3.  28.  .  ^MZ  ^  A^ 

29.    Va6-i-3V6c.  30.    m-^Vn. 

217.  Multiplication    of    polynomials    containing    surds. 

The  product  of  two  polynomials  involving  surds  is  found 
by  direct  application  of  the  distributive  law  for  multipli- 
cation ;  the  operation  differs  in  no  respect,  from  that 
required  in  the  multiplication  of  rational  integral  poly- 
nomials. Each  term  of  the  resulting  product  must  be  ex- 
pressed in  its  simplest  form. 

ILLUSTRATIVE  EXAMPLE 

Multiply  2  4-  3  V5  -  2  V2  by  3  V5  -  4  V2. 

Solution. 

2  +  3V5-2V2 
3V5_4V2 
6\/5  +  45-6VlO^ 

+  16  -  12V10  -  8\/2 
6V^  +  61-18VlO-8\/2. 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     283 

Remark.  Observe  that  it  is  necessary  to  simplify  the  terms  of 
each  partial  product  in  order  that  similar  surds  may  be  written  in 
the  same  column. 

EXERCISE  104 

Simplify  each  of  the  following  expressions  : 

1.  (2  +  V5)(2-V5).  2.    (a  +  2V3)(2-fV3). 

3.  (3-2V2  +  V3)(V6+V2). 

4.  (aV5  +  6V^)(Va+ V6). 

5.  («+ V5-f  \/^)(a- V"^+ Vc). 

6.  (V^-hV3)2. 

7.  (VS_V5)2-(Va  +  V^)2. 

8.  (aVb-b^/ay. 

9.  Find  the  product  of  3V2  -  V3  and  V2  +  3V3. 

10.  Find  the  product  of  9  V3  +  2  and  2  V3  H-  9. 

11.  Find  the  product  of  V2  -  V3  +  1  and  V2  +  V  3  + 1. 

Find  the  square  of : 

12.  V3-1.  13.    2  +  V2. 
14.    3V3-V2.  15.    \^-2V2. 

Find  the  cube  of : 

16.    V3-1.  17.    2  +  V2. 

18.    3V3-V2.  19.    V5-2V2. 

20.  Simplify  (1  +  3V3)(9  -  V2)(V2  +  V3)(9  +  V2) 
(3\/3-l)(V2-v'3). 

218.  Conjugate  surds.  Two  binomial  quadratic  surd 
expressions  which  differ  in  sign  of  a  surd  term  only  are 
called  conjugate  surds. 

Thus,  1  +  V2  and  1  -  V2,  also  V2  +  V3  and  V2  -  V3  are  conju- 
gate  surd  expressions. 


284  ELEMENTARY  ALGEBRA 

219.  Product  of  conjugate  surds.  The  product  of  two 
conjugate  surds  is  rationaL 

Thus,  (Va  +  Vb)(Va  -  y/b)  =  (V^y  -  (\/&)2  =  a  -  b. 

Note.  When  the  product  of  two  expressions  which  contain  irra- 
tional numbers  is  rational,  either  expression  is  called  a  rationalizing 
factor  of  the  other. 

ILLUSTRATIVE  EXAMPLE 

Rationalize  the  denominator  of  the  fraction 

1+V2 
V5  +  V2' 

Solution.  Since  the  product  of  two  conjugate  surds  is  rational 
[section  219],  we  multiply  the  numerator  and  denominator  of  the  frac- 
tion by  a  conjugate  of  the  denominator.     Then  we  have, 

l-fV2    ^    (l  +  V2)(\/5-V2) 
V5-fV2      (>/5-fV2)(V5-V2) 

^  V5  -I-  Vio  _  V2  -  2 
3  ' 

EXERCISE  105 

Rationalize  the  denominator  of : 


2. 


14.V2  2-V2  2  +  V2 


V3-V2  V3-V2  2-Va; 

X  .     1-V5 


m 


n  +  Vw  -y/x  +  V^  1  +  V5 

2  +  V3  „     V3+V2  „     yg  +  v^ 

•    2-V3  V3-V2  Va-V5 

13.    _^2_  ^^     4±V3  ^3       ^-2_ 
V5-f-V7                  4~V3  V12  +  V8 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     285 
Find  the  sum  of : 
16.        _^       +      ^      .  17.       9V3  2V2 


V3  +  1      V3-1  *    2V3-f3     3V2  +  2 

Suggestion.     Rationalize  the  denominators. 

18.     4.-^— L.  19.    -2=4-     '  ^ 


V27     V48      Vl2  V60     Vl5     Vl35 

Rationalize  the  denominator  of : 

4  Suggestion.     Multiply  both  terms  by 


20 


2-(-V3-fV5'  2+V3-V5. 


21.     12 ^  33^  1 


V2  4-V3+V5  V2  +  V5-V8 

220.  Division  by  polynomials  containing  surds.  To 
divide  an  expression  by  a  polynomial  containing  one  or 
more  surds,  the  dividend  should  be  written  as  the  nu- 
merator and  the  divisor  as  the  denominator  of  a  fraction, 
which  should  be  transformed  into  an  equivalent  fraction 
with  a  rational  denominator. 

EXERCISE   106 

Perform  the  indicated  divisions  in  the  following  : 

1  «        V2  ,     V3-V2 

3. 


V2-1  V2-I-1  V3-f-V2 

1  ,     2-hV3  ^        3-V5 


V5-V3  1  +  V3  V10-V6 

7.  Find  the  quotient  of  (V2  +  V6)-5-(l -}- V3)  with- 
out rationalizing  the  denominator. 

8.  Mention  at  sight  the  numerical  value  of  3V2+y ^ 

1+V2 


286  ELEMENTARY  ALGEBRA 

V2  +  V4-v^ 


H.V2-V3 


9.    State  at  sight  the  value  of 

10.   Simplify     V3-f'^^~^» 
^    ^  14-V3 

"•  -p>«'  (^T-e-^T- 

12.  Divide  (3  -  V2)2  + 1  by  3  -  V2. 

13.  Divide  (V3  +  V2)2+l  by  V3-hV2. 

14.  Find  the  value  of  ^-^^^-^^-^  ^hen  a;  =  1  +  V2. 

X 

221.  Fractional  and  negative  exponents.  The  following 
identities  from  section  198  had  a  meaning  only  when  the 
exponents  were  positive  integers. 

«"».«"  =  «"*+".  (I)  (a5)'"  =  aH"^.  (II) 

(«"»)«  =  («")»«  =  a"*".     (Ill)  (a"^5")p  =  a*"Pft"p.        (IV) 

In  section  62  the  identity 

—  =  a»»-",  in  which  m>n  (V) 

was  shown  to  result  directly  from  the  definition  of  divi- 
sion, and  in  section  63  a  meaning  was  given  to  the 
expression  a9,  where  a  denotes  any  number  different  from  0. 
That  is,  it  was  shown  that, 

It  is  convenient  to  extend  the  meaning  of  the  word 
exponent  so  that  this  term  shall  include,  in  addition  to 
positive  integers,  all  other  rational  numbers. 

It  is  evidently  desirable  that  all  exponents  should  com- 
bine according  to  the  same  laws,  and  hence  to  define 
negative  and  fractional  exponents  so  that  the  identities  of 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     287 

section  198  may  be  satisfied  for  all  rational  values  of  w, 
w,  and  p.  We  shall  proceed  to  show  that  such  definitions 
can  be  given.  In  what  follows,  any  base,  as  a,  is  supposed 
to  be  different  from  0. 

222.  Definition  of  a'"".  If  the  identity  «»".«»=  a^+", 
in  which  m  and  n  are  positive  integers,  is  also  an  identity 
when  n  is  replaced  by  —  w,  we  shall  have 

^m  .  ^-m  __  ^m+(-m)  _  ^0  _  J^ 

We  therefore  define  a~^  by  the  equation  a^  .  a~^  —  1 ; 
hence,  a~^  and  al^  are  reciprocals  ;  that  is, 

cr'"  =  —  ;  also,  O"  = 

It 

223.  Definition  of  af.  If  the  identity  («*")*»  =  a*"",  in 
which  m  and  n  are  any  positive  integers,  is  also  an  identity 

when  m  is  replaced  by  ^,  where  p  and  q  are  positive 
integers,  we  shall  have,  by  taking  n  equal  to  ^, 

p 
Therefore  a^  must  represent  a  number  whose  qth  power 

is  equal  to  a^.     Now  the  principal  qth  root  of  a^  is  such  a 

p 
number.     We    therefore    define   the    symbol   a^   by  the 

equation 

0?  =  -?/^.  (I) 

In  this  identity,  Va^  represents  the  principal  5th  root  of 
aP.     In  particular,  by  making  j3  equal  to  1, 

a^  =  Va,  (II) 

Since  p  and  q  in  identity  (I)  represent  any  positive 
integers,  it  is  admissible  to  change  them  in  this  identity  to 


288  ELEMENTARY  ALGEBRA 

pm  and  qm^  respectively;  we  then  have  from  identity  (I) 

pm  

But,  V^=^^  [§209  (II)] 

p 

pm  p 

a'""=^a'^.  (Ill) 

p 
From  identity  (III)  it  is  evident  that  the  value  of  a^  is 

not  changed  when  £  is  replaced  by  an  equivalent  fraction. 

9 

p 

Note.  In  the  expression  a',  the  numerator  p  indicates  a  power 
and  the  denominator  q  a  principal  root.  In  some  cases  it  may  be 
desirable  to  take  the  pth.  power  of  the  principal  ^th  root  of  a  and  in 
others  to  take  the  principal  ^th  root  of  the  joth  power  of  a.  That, 
in  both  cases,  the  results  are  the  same  is  evident  from  the  known 
identity,  section  209,  IV, 

5 

Thus,  in  simplifying  8^,  either  of  two  solutions  may  be  given,  the 
first  being  preferable. 

Solution  1.     8'  =  (8^)6  =  (2)5  =  32. 

Solution  2.     8^  =  (85)*  =  K^^yi^  =  (2i6)*  =  2^  =  32. 

In  simplifying  2^  it  is  evident  that  the  method  followed  in  solution 
2  is  preferable ;  that  is, 

2^  =  v^22  =  H. 
In  simplifying  2^  it  is  preferable  to  proceed  as  follows : 
2^  =  2^"^^  =2  X  2^  =  2v^4. 

_p  _p 

224.  Definition  of  a  '^ .  From  section  222,  a  *  must  be 
defined  by  the  identity 

-^       1 

a  </  = 

p 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     289 

225.  It  will  be  found  that  with  the  foregoing  definitions 
of  negative  and  fractional  exponents,  identities  (I),  (II), 
(III),  (IV),  and  (V),  section  221,  now  hold  for  aU 
rational  values  of  w,  n^  and  p.  We  shall  agree  to  enlarge 
the  meaning  of  the  word  "  power  "  in  accordance  with  these 
definitions.  For  example,  by  the  expressions  the  "  one- 
third  power   of   a"  or  "a  to   the   one-third  power,"  we 

shall  mean  the  number  a^;  that  is,  -^a. 

ILLUSTRATIVE  EXAMPLES 

1.    Simplify     a^  xa^. 


Solution.                      a^  X  a^  =  a^'^^  =  a^. 
2.    Simplify     ^. 

[§§225,221(1)] 

Solution.                         ^=x^~^=x. 

[§221(V)] 

JC 

3.    Simplify     (^ah^y. 

Solution.                      {ah^y  =  (a*)«(6^)6. 

[§  221  (H)] 
[§221  (III)] 

4.    Write  \-^  without  using  the  radical  sign. 

Solution.                         ^}^  ={  ^\   • 
^2^y      \x»yJ 

[§  223  (11)] 

5.    Simplify     mhi-^  -;-  m~%*. 

Solution.             ?n^-^  -  m-8n*  =  !^  ^  ^  =  ^. 

n*      m*      n« 

[§222] 

6.   Simplify     Q 

Solution.        (M^Jlii-    1    _    1    = 

xy 

[Exercise  94,  problem  33] 


290  ELEMENTARY  ALGEBRA 


'•  «-^'"'  ©' 


Solution.         (S?^]^  =  («^^^)^  =  ^  =  MJJ^ 

[Exercise  94,  problem  33] 

8.  Simplify     ^'-^. 

Solution.         ^=  ( J)*=(,-i)i^(,^)i  ^  ,i  ^  e,,. 

9.  Simplify  (^-LW^X^^). 
Solution  i'yP-)  (^  ^  ^^)  =  (4-)  ^V 

=^ 


a-iV^ 


=2t 


ia:T^ 


Z  X 


10.    Multiply  a;^  +  x^y^  +  ?/^  by  a;*  —  y^. 
Solution.  x^  +  x*y^  +  y^ 


J- 

,v* 

J  + 

xV 

-,v* 

.i 

-y* 

POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     291 
11.   Divide  x  —  yhy  x*  —  y^. 

Solution.  x^  —  y^)x  -  y\x^  -\-  x^y^  +  y* 

X  —  x^y^ 

2     1 

x^y'^  -  y 

x^y  -y 


X,.    Simplify  1  V^.^H.-M)» 

x~^  —  y~^  X  —  y 

Solution.     — -, :  =  T — =•  =  — — 

x~^  —  y~^     1  _  1     y  —  X 

X      y 

\X^J  X^ 


X^     ^ 

y 


1  y/x^y-^{x  2^2)8  _     a:y         y^    x 

p-i_  y-i  a-  _  2,  ~  y  -  X       x  -  I 


^     xy      ^      xy    ^    xy  xy    ^  q^ 

y -  X      X- y      y -X     y - x 


Check.  Let  a:  =  4  and  y  =  1]    then, 

=  -1  +  1  =  0. 


292  ELEMENTARY  ALGEBRA 

EXERCISE  107 

(Solve  as  many  as  possible  at  sight.) 

Write    without  negative  or  fractional  exponents,  and 
simplify  when  possible  : 


1. 

a-\ 

Thus,  a-2  =  ^. 
a* 

2. 

4i 

Thus,  4^  =  +  V4  =  2. 

3. 

8l 

Thus,  8^=(\/8)2  =  22  =  4. 

4. 

8-^. 

T.^8-*=J=i. 

5.    2-2. 

6. 

9*. 

7.    16i                 8. 

4^. 

9.    x-^. 

10. 

9i 

11.    16^.               12. 

27t. 

13.    a;-". 

14. 

8-1 

15.     25-i              16. 

64-*. 

17.  What  verbal  statement  is  expressed  in  symbols  by 
the  formula  a*"a"  =  «"*+"  ? 

18.  What  verbal  statement  is  expressed  in  symbols  by 

the  formula  —  =  a*"""  ? 
a" 

19.  What  verbal  statement  is  expressed  in  symbols  by 
the  formula  («"»)"=  a*""  ? 

Express  without  fractional  exponents  : 

20.  ai                          21.    rpi                          22.  y^. 
23.    3a^.                      24.    2a;i                      25.  4yi 
26.    (3a)i                 27.    (2  a;)*.                 28.  (4y)*. 
29.    a^.                       30.    2ai                     31.  (2  a)*. 
32.    a^a;.                      33.    a^xy^.                  34.  2a^a:i 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     293 


1^ 
35.    a"*. 

36. 

n+1 

37. 

2y. 

38.    2hxl 

m 

39. 

a-b 

40. 

m 

ax^ 
3   * 

41.     ^^". 
44.    a^  +  bhK 

42. 

Cx-;,-)i. 

43. 

2a¥. 

45. 

©' 

46. 

m— n     Bi+n 

Express  without  the  radical  signs : 

47.     </i. 

48. 

</a. 

49. 

^?^. 

50.    4^. 

51. 

3Vi. 

52. 

5^J. 

53.     ^/Sa. 

54. 

V7  2:. 

55. 

^5"c. 

56.     V42:3. 

57. 

V2a3. 

58. 

2VP. 

59.     v21xK 

60. 
63. 

aVa;. 
Vx, 

61. 
64. 

as/he. 

62.     7Vi^. 

V2  a'»+». 

65.    V3a:3. 

66. 

e'</4x^, 

V2 

67. 

""■v/a-+». 

68.     |VF. 

69. 

70. 

^(a+4)» 

71.    SVa^K 

72. 

V2a:-f  Vy2. 

73. 

74.    -■'^2--'''-^ 

V 

i 

75.    ->/^V6^ 
77.     ^V^. 

y?. 

76.    aV^  ""'"■v/a;"**-*' 

226.   Square  root  of  a  binomial  surd  expression.     The 

square  of  a  binomial  surd  expression  of  the  form  a  +  V3 
is  itself  a  binomial  surd  expression  of  the  same  form. 
Thus,  (2  -\/5)2  =  9  -  4V5  and  (a  +  ^ly  =  a^  +  6  +  2a\/6. 

When  an  expression  in  the  form  of  a  -f-  V3  is  a  perfect 


294  ELEMENTARY  ALGEBRA 

square,  its  square  root  may  usually  be  found  by  inspec- 
tion. The  method  may  be  seen  in  the  solutions  of  the 
following : 

ILLUSTRATIVE  EXAMPLES 

1.  Find  the  square  root  of  16  -h  6V7. 

Solution.  16  +  6  V7  =  16  +  2v'63 

=  9  -f  2V9  V7  +  7 

=  32  +  2  X  3  X  V7+(V7)2 

=  (3+V7)2. 

.-.  V16  +  6V7  =  V(3  +  V7)2  =  3  +  V7. 

2.  Find  the  square  root  of  6  —  Vll. 
Solution.  6  -  Vll  =  6  -  2  V^ 

=  ¥  -  2  V-V-  vi  +  i 
=(V5y-2V-y:V|+(Vjy 


.-.  Ve-Vii  =  V(  v^  -  Vly 


3. 


Show  that  V  6  +  V35  +  V 6  _  V35  =  Vl4. 


Solution.         Ve  +  V35  =  V  6  +  2  Vy  =  V|  +  V| 
V6-V35=V6-2V^  =  V|-V| 


Sum  =  2  Vj  =  vTi. 

227.  From  the  solution  of  the  foregoing  illustrative 
examples  we  may  state  the  following  : 

Rule.  A  binomial  surd  expression  of  the  form  a-\-  2V6, 
where  a  and  h  are  rational  numbers^  is  a  perfect  square  when  b 
(the  number  under  the  radical  sign^  is  the  product  of  two  fac- 
tors whose  sum  equals  a,  the  rational  term.     In  this  case,  the 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     295 

square  root  of  the  binomial  surd  expression  is  eqyxil  to  the 
sum  or  difference  of  the  square  roots  of  the  two  factors  of 
the  number. 

Remark.     Any  binomial  surd  expression  of  the  form  a  +  Vft  can 

be  written  a  +  2\-' 

228.    (1)  A  quadratic  surd  cannot  he  equal  to  the  algebraic  sum  of  a 
rational  number  and  a  quadratic  surd. 

For,  if  possible,  let  Va  =  &  +  Vc ; 

then  by  squaring,  a  =  b^  -\-  c  -\-2  bVc. 

and  Vc=  «-^'-^ 


2b 

Since  Vc  is  a  quadratic  surd,  it  is  an  irrational  number  and  can- 
not be  equal  to  the  rational  number"     oa~  ^ '    Therefore,  the  assump- 

^  0 

tion  that  Va  =  6  +  y/c  is  false. 

(2)  A  binomial  surd  expression  of  the  form  a  +  y/b  cannot  be  equal 
to  another  expression  x  +  y/y  of  the  same  form  (where  a  and  x  denote 
rational  numbers)  unless  a  =  x  and  Vb  =  y/y. 

For,  if  possible,  let         a  +'"\/6  =  a:  +  Vy ; 
then,  Vb  =(x  —  d)-\-  y/y. 

By  (1)  this  equation  can  be  true  only  when  a:  —  a  =  0,  in  which 
case  y/b  =  y/y. 

EXERCISE  108 

Find,  in  surd  form,  the  square  root  of  : 
1.  16-6V7.        2.  6  +  VTT.  3.  7-hV48. 

4.  32-V700.       5.  12-6V3.  6.  8-I-2V7. 

1 


7.    Simplify 


Vs-i-Vs 


9.    Simplify  Vn  + 12  V2. 


296  ELEMENTARY  ALGEBRA 

EXERCISE  109.    REVIEW 

(Solve  as  many  as  possible  at  sight.) 
Write  without  negative  exponents  : 


1. 

ar\ 

2. 

a%-^. 

3. 

a-^c. 

4. 

x'^yr^. 

5. 

a-^b^. 

6. 

2a-%'\ 

7. 

Sa-H. 

8. 

5-1  a. 

9. 

iaby\ 

10. 

5a-\ 

11. 

3a6-8. 

12. 

x^r^. 

13. 

a^c-K 

14. 

a«6-». 

15. 

3-32^. 

16. 

6-1 
3 

17. 

aT'^xif^, 

18. 

(5a-i)-i. 

19 

1 
a-2 

20. 

1 

a;-i 

21. 

1 

X«7. 

(2^)-i 

22. 

a 

23. 

a 

24. 

a 

Sxy-' 

3(xy)-> 

(Zxyy-^ 

25. 

a-i 

26. 

a;«-iy-»-l. 
(-a)-3. 

27. 

7fl~^, 

(3=^y)-' 

1 
_1 

a  « 

28. 
31. 
34. 

ia  +  x)-^ 

2-8 

16-1 
(-  «)-*• 

29. 
32. 
35. 

30. 

33. 
36. 

37. 

©"■ 

38. 

<!)"■■ 

39. 

a-^b-^c 

Aft 

41 

(3  aby 

-8 

n\}. 

•    (3a3)-3(2  6*)"^ 

42. 

1 

-^ 

-1 

-c-i 

a-i  +  6-1 

AA. 

1 

-^ 

Vm* 

a-i-1 

POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     297 

Perform  the  indicated  operations  in  examples  46-51 : 

46.    a*  X  a^,  47.    4  a~^ -s-  2 x'K 

48.    aT^V'^.  49.    (a*)^-5-(a:^)2. 

50.    (a^6"^)i  51.    (a"^)»x(a^)*x(a~^)-^. 

52.  Find  the  product  of  a^,  a^  a^  a^,  a",  and  a\ 

53.  Find  the  product  of  rnJ^  m*^  m",  m*,  m~^  and  w"^. 

54.  Express  9m^n~^  -i-  15n^p~^  x  20jt?277j-3  with  positive 
exponents,  perform  the  indicated  operations,  and  reduce 
the  result  to  simplest  form. 

55.  Express  the  following  with  positive  exponents, 
add,  and  simplify  the  result : 

x~\yz~^  —  y~^z)-\-  yr\zx~^  —  z~\c)-\-  z-\xy-^  —  x~^y). 

Simplify  the  following : 


56. 

9l 

57. 

4i 

58. 

64*. 

59. 

25i 

60. 

-27i 

61. 

(-27)*. 

62. 

(27)-*. 

63. 

(-27)-*. 

64. 

32*. 

65, 

100-^. 

66. 

2 
4-2 

67. 

256*. 

CO 

(-i)-^. 

69. 

(A%)"^. 

70. 

20 

68. 

(-l)-7 

71. 

516 

512* 

72. 

4-^  X  36i 

73. 

4-2^5-3 

74.  64"^ -64"* +(-64)^ +  64*. 

75.  3-*  X  Vsi  X  5-^  X  V5. 

76.  (a;-2)3.  77.    (^2)-8.  78.    (a;-i)-i. 

79.  (a3)*.  80.    V^.  81.    (64*)2. 


298  ELEMENTARY  ALGEBRA 


1     .    1 


82.    —-h  —  .  83.    (64a-6)l  84.    (a^)-2. 

85.    (-a8)6.  86.    (-a3m)4,  g^     (^a'^h'^c^^^. 

as.   lla-^pp.     a.  ©".  so.  (1^)-^- 

91.    8"^  X  16^  X  20.  92.     — -i— -. 

2-1  +  3-1 

93.     (a +  6)0.  94, 


95. 


97.    2«-i  x8«+1h-162.  98.     ^ — ^. 

99.    (a°)".  100.    V«~i^^a*. 

101.   (af-^yhy^-^y-y.  102.   [(e^-e-*)^  +  4]«. 

103.  (a;^  +  a:^  +  1)  (a;*  -  a;^  +  1). 

104.  aizu^  +  ^l±L^.  105.    [V^Fh^]*. 
a^  _  ji      a^  +  h^ 

1  1 

106.  [(^0"']    ^-5-[(«"0'']    *• 

107.  (a^  -  ah^  +  a^6  -  6^)  ^  (a*  -  6^). 

108.  ix-^y-^  +  xy^y,  109.    [3V5=V]-*«. 

110.  (-V?)io  +  (2V3)«-a;-Y--?^y. 

111.  (a;  +  2a;^  +  l)i 

112.  [a;-2a;^  +  3a^^~2a;^  +  l]*. 


POWERS,  ROOTS,  RADICALS,  AND  EXPONENTS     299 
Clear  of  negative  exponents  and  simplify : 

113.  .^  ♦  114. 


m~^ -H  ^~^  ---       (^m—n)~^ 


Arrange  according  to  descending  powers  of  x : 

115.  x^  -  2x^-^4  x^  -{-  x^. 

116.  Since  x^  —  i/^  =  (a;^)^  —  (?/^)2,  express  x^  —  3/^  as  a 
product  of  the  sum  and  difference  of  the  same  two 
numbers. 

117.  Simplify     — i-^+      1  2  a^ 


a^_l      a^  +  l      a-1 

118.  Evaluate  a/«2  .  ^/^b  .  Vo^"  ^  ^^  when  a  =  3. 

119.  Evaluate  5"  •  25"+i .  125-"+i  -r-  5^. 

120.  Simplify  Jl  _  [1  _  (1  _  a^yiyi^i, 

121.  Evaluate  (243)^  + (243)i 

122.  Simplify  (\2V2V2Vl)^- 

123.  Simplify  7/~^[(x  +  V^)3  -(x-  V^)^]. 

124.  Which  is  the  greater  \/27  or  ^16  ? 

125.  Simplify  (a^"^)'  -;-  (a:"^)*  -5-  (2:"^)^  h-  (a;"^)i 


a«-6« 


126.  Simplify  (x^^+f>^'  -^  a:(«-*>')  ^"* 

127.  Simplify ^~^ 


CHAPTER   XI 
INVOLUTION   AND   EVOLUTION 

Involution.     The  operation  of  raising  an  expres- 
sion to  any  positive  integral  power  is  called  involution. 

Remark.  The  involution  of  monomials  has  been  explained  in 
section  198.  We  shall  now  consider  the  involution  of  a  binomial. 
Since  by  grouping  the  terms  of  any  polynomial  it  may  be  expressed 
as  a  binomial,  the  involution  of  a  binomial  is  an  important  topic  of 
algebra. 

230.  Binomial  expansion.  The  process  of  raising  a  bi- 
nomial to  a  power  is  called  expanding  the  binomial,  and  the 
result  of  the  operation  is  called  a  binomial  expansion. 

Thus,  by  expanding  (a  +  b)^  we  obtain  a^  -\- 2  ab  -{•  b%  which  is 
called  the  expansion  of  (a  +  by. 

231.  The  binomial  formula.  By  actual  multiplication 
we  obtain  the  following  expansions  : 

(a  -f.  5)2  =  «2  4.  2  a6  +  62. 

(a  +  by  =  a3  +  3  a^  +  3  a62  4  ^,3. 

(a  +  6)4  =  a*  +  4  a%  -h  6  a2^>2  +  4  ^J3  +  ^4. 

(a  +  6)6  =  a^  -j-  5  a'^b  +  10  a^I>^  +  10  a^^  +  5  aft*  +  b^ 

If  we  examine  the  above  expansions  we  arrive  at  the 
following  important  conclusions  which  we  here  assume 
are  true  when  a  binomial  is  raised  to  any  positive  integral 
power: 

1.  77ie  first  term  in  the  expansion  is  a",  where  a  is  the  first 
term  and  n  is  the  exponent  of  the  binomial.     The  last  term  is 

300 


m^ 


Isaac  Newton  (1642-1727)  was  perhaps  the  greatest  mathe- 
matician of  all  time.  He  is  best  known  as  the  discoverer  of  the 
laws  of  gravitation.  In  algebra  he  discovered  the  binomial  theorem 
and  wrote  extensively  on  the  theory  of  equations.  His  great  work 
"  Philosophiae  Naturalis  Principia  Mathematica"  appeared  in 
1686-87. 


INVOLUTION  AND  EVOLUTION  301 

6",  where  h  is  the  second  term  of  the  binomial.     The  number 
of  terms  is  n  +  1. 

2.  TTie  sum  of  the  exponents  of  a  and  b  in  any  term  is  n. 
The  exponent  of  b  in  the  first  term  is  zero  and  that  of  a  in  the 
last  term  is  zero.  The  exponent  of  a  in  the  second  and  fol- 
lowing terms  is  one  less  than  in  the  preceding  term.  Hence, 
the  exponent  of  b  in  any  term  is  one  greater  than  in  the  pre- 
ceding term. 

3.  The  coefficient  of  any  term  is  obtained  from  the  coeffi- 
cient of  the  preceding  term  by  multiplying  that  coefficient  by 
the  exponent  of  a  in  that  term  and  dividing  the  resulting 
product  by  ofne  unit  more  than  the  exponent  of  b  in  that  term, 

ILLUSTRATIVE  EXAMPLES 

1.  Expand  (2  +  xy. 

Solution.     Since  (a  +  6)8  =  a^  +  3  a^ft  +  3  ah^  -I-  h\ 
therefore,  (2  +  a:)^  =  28  +  3  •  22  .  a;  +  3  •  2  •  ar^  +  a:« 

=  8  + 12  a; +6x2+3:8. 

2.  Expand  (a  —  2  ?/)*. 

Solution.     Since  (a  -  by  =  a^  -  ^a%  +  Q  a^h"^  -  4  aft8  +  h\ 
.'.  (a  -  2  i/)4  =  rt*  -  4  a8(2  2/)  +  6  a\2  yY  -  4  a(2  yY  +  (2  y)* 
=  a*  -  8  a8y  +  24  02^2  _  32  ay^  +  16  y*. 


3.    Expand  C^-^Y 


Solution. 

Since  (a  -  hy  =  a^  -  Q  a%  +  15  a*62  _  20  a%^  +  15  a%^-  6  aft«+  6«, 

^1      6  &  J  15  />2     20  68  ^  15  &^     Q¥     h^ 

/jB         n^n  n^r^  n^r*^  n^n^  n/^o  /.6 


302  ELEMENTARY  ALGEBRA 


EXERCISE  110 

(Solve  as  many  as  possible  at  sight.) 

1. 

(x  4-  2/)^ 

2.    Cx-yy. 

3. 

(a  +  hy. 

4. 

(m  —  ny. 

5.  cx-iy. 

6. 

(2  -  x-)\ 

7. 

(a  +  hy. 

8.    {x-yy. 

9. 

(X^  -  1)3. 

10. 

(3  a  -  by. 

11.    (3a+2J)3. 

12. 

(5*- 3^)3. 

13. 

(2  m  -  ny. 

14.    (2  a2  -  1)4. 

15. 

(2-3  nfiy. 

16. 

(3  2^  +  2  fy. 

17.    (2  a  4-  ^>)^ 

18. 

(1  -  ay. 

19. 

(_^2_  2^2)6. 

20.    (r2-2)7. 

21. 

(mhfi  -  1)». 

22. 

(1^2+1)5. 

23.     (1:^2-  1^2)6 

.    24. 

(J^^+3)». 

25. 
28. 

Qm-hiny. 

»•  g-:)* 

27. 

(l-i)' 

/ 
29.       1  + 

v 

a-     - 

S 

..)' 

232.  Evolution.  The  operation  of  extracting  an  indi- 
cated root  is  called  evolution. 

233.  Square  root.  In  section  88  a  square  root  of  a 
number  was  defined,  and  it  was  shown  that  any  number 
has  two  square  roots  of  the  same  absolute  value  but  with 
opposite  signs. 

From  section  92  we  have  (a  -k-hy=  a^-\-  2  a5  4-  62;  there- 
fore,   

V«2  +  2ah  +  l^==  ±  (a  +  V). 

From  section  209  (V),  we  have 


therefore, 

J2~vP'"      * 


vfiT=V^=-=^-" 


INVOLUTION  AND  EVOLUTION  303 

EXERCISE   111 

Find  the  following  square  roots  by  inspection: 
1.    Vl.  2.    Vl6^.  3.    V9^2p. 


4.    V625  2:y.  5.    +Vl6rc^2^.  6.    -  V324  a*. 

7.    +V289^.  8.    -VT02l«V. 


9.    Va2-2a6+^>2.  10.    _Va2  +  4a  +  4. 

11.    V42:2_4^_^i,  12.    Va:2  _  6  ^  +  9. 


13.  Va:V-2a:«/  +  l.  14.  -\-V9 x^-hl2x^  +  4  f. 

15.  -Vl6a2  +  24a^.H-9R  le.  +  V64  x  81  a%^(^. 

17.  +  V4  X  25  X  256  a^js^ioye. 

18.  -V^\  19.     +  \/a2"+2^  20.     4-V«2p^. 


21. 


23. 


25. 


J?^.  24      x/? 


2_2a6  +  52 


729  ^a2^2a^  +  ^ 

\a;2_io^  +  25'  •        ^92:2-18a:  +  9  ' 


234.  Square  root  of  a  trinomiaL  The  positive  square 
root  of  any  trinomial  which  is  a  perfect  square  may  be 
found  by  inspection. 


Thus,  +  Va2  -{-2ab  -\-b^  =  a-\-b. 

The  actual  work  of  finding  the  square  root  of  a2-f-  2  ab-\-b^ 
may  be  arranged  as  follows: 


a2  +  2  a&  +  &2  \a-\.b 
a2 


trial  divisor  2a 

complete  divisor  2a  +  b 


2ab  +  b^ 
2ab  +  b^ 


0 
The  explanation  of  the  method  is  as  follows: 


304  ELEMENTARY  ALGEBRA 

1.  We  find  the  square  root  of  the  first  term  (which  is  the  first  term 
of  the  result)  and  subtract  its  square  from  the  trinomial,  obtaining 
2  a6  +  b\ 

2.  The  second  term  h  of  the  result  may  be  found  by  dividing  2  ah 
by  2  a.  We  call  2  a  the  trial  divisor.  The  trial  f/irisor  /.s  double  the  part, 
a,  of  the  root  already  found.  After  h,  the  second  term  of  the  result,  has 
been  found,  we  add  it  to  the  trial  divisor,  obtaining  2  a  -\-  b,  the 
complete  divisor. 

3.  Multiply  the  complete  divisor  2  a  -\-  bhj  b  and  subtract. 

235.  Square  root  of  any  polynomial.  The  process  em- 
ployed in  finding  the  square  root  of  a^  -{-2ab  + 1^  is 
applicable  in  finding  the  square  root  of  any  polynomial. 

ILLUSTRATION 

Extract  the  square  root  oix^  —  2  x^i/  +  Sx^i/^  —  2  xi/^  4-  ^. 
x^-2x^y  +  S  x^y^ -2xy^+ y^ \x^-xy-\-y^ 


1st  trial  divisor,  2(x^)  =2x^ 
1st  complete  divisor,    2x^  —  xy 
2d  trial  divisor,  2(x^  -xy)  =  2x^— 2  xy 
2d  complete  divisor,  2x'^—2xy-\-  y^ 


2  x^y  +  3  x^y^ — 2  xy^ + y*,  1st  remainder 

2xhi+    a;V 


-f-  2  x'^y^ — 2  xy^ + «/*,  2d  remainder 


0, 3d  remainder 


Explanation.  1.  The  square  root  of  x^  is  x^  (the  first  term  of 
the  result).  The  first  trial  divisor  is  2x^  (double  the  part  of  the 
root  already  found).  The  first  term  in  the  first  remainder  divided 
by  the  trial  divisor  is  —  xy  (the  second  term  of  the  result),  and  the 
first  complete  divisor  is  2x^  —  xy.  Here  a  =  x\  2ab  =  —  2  x^y, 
2  a  -^  b  =  2  x^  —  xy.  Multiplying  2  x^  —  xy  hj  —  xy  and  subtracting, 
we  obtain  the  second  remainder;  this  corresponds  to  multiplying 
2a  +  bhj  b  and  subtracting. 

2.  The  second  remainder  is  the  result  of  subtracting  the  square  of 
the  part  of  the  root  already  found  (x^  —  xy)^  from  the  given  poly- 
nomial. By  taking  a  =  x^  —  xy,  the  second  trial  divisor  is  found  to 
be  2(x^  —  xy)=:  2x^  —  2xy.  The  first  term  of  the  second  remainder 
divided  by  the  first  term  of  the  second  trial  divisor  is  y^  (the  third 
term  of  the  result),  and  the  second  complete  divisor  is  2  x*  —  2  xy  +  y\ 


INVOLUTION  AND  EVOLUTION  305 

We  now  multiply  2  x^  —  2  xy  +  y^hj y^ and  subtract.  Here  a  =  x^  —  ary, 
2a  =  2x^-2xy,b  =  y\B,nd2a  +  h  =  2x^-2xy-\-yK 

3.  The  third  remainder,  0,  is  the  result  of  subtracting  the  square 
of  the  part  of  the  root  already  found  (x^  —  xy  +  y^Yi  from  the  given 
polynomial.  Hence,  the  given  polynomial  is  the  square  of  x^  —  xy  -\-  y^, 
and  x^  —  xy  -{-  y^  is  the  required  square  root. 

Remark.  The  polynomial  should  be  arranged  according  to  the 
powers  of  some  one  letter  before  the  work  of  extracting  the  square 
root  is  begun. 

EXERCISE    112 

Find  the  square  root  of  the  expressions  in  examples  1-9. 

1.  9a^-Qx-\-l. 

2.  4:2^-12a^-{-lSx^-6x-hl, 

3.  25a^-20a^  +  Ma^-12x-{-9, 

4.      3^-\-22^-Sx^-4:X-\-4:. 

5.  3^-4:a^  +  5x^-2x-\-\. 

6.  x^-2x-\-S---h-. 

X      x^ 

7.  7^-2x^y-\-b T^y'^ -  14 2:^^3+14 a^»^-20 xf-{-2b /. 

8.  x^+2x^-2a^-^x^-2x-{-l, 

9.  x^-\-2x^-^l  +  2x^  +  ^x^. 

10.  Show  that  \  +  x^^(2x^- 4)a?»  +  (a;+  l)(a;  +  3)  is  a 
perfect  square. 

11.  For  what  value  of  m  is  the  expression  4  a;*  —  12  a^y 
+  mx^y'^  —  6  xy^  +  y^  ^  perfect  square  ? 

Extract  the  square  root  of  the  expressions  contained  in 
examples  12  to  26  inclusive. 

12.  aV+h'^y'^+l  +  2ahxy-ir2ax-^2hy. 

13.  x^-Qmx^-\-  13 m^x^ - 20 m^a^-{- 16 m'^a^- 6 m^x-\-m^. 

14.  (p -  qy-  2(jt?2  +  fxp  -  qy+Kp"^  +  ^0. 


306  ELEMENTARY  ALGEBRA 

15.  aP-.4x-\---  —  'hlO. 

of       X 

-^     m^  .  n^  .  2m  .  ^n  .  ^ 

16.  -5  +  — ^  H 1 1-  o. 

fir      m^       n        m 

17.  a2j-2  4-  J2<?-2  +  c2a-2  +  2  a<?-i  +  2  <?6-i  +  2  har\ 

18.  a3  4-2a2  +  a  +  l-2a*(a  +  l). 

19.  (a^  +  «2  4-  2  a^  +  2  a  +  a^  +  1)(«*  -  l)(a  -  1). 

20.  if  +  $2)  (r2  +  «2)  _  (^^  _  ^y.)2. 

21.  -^^+^-.^-2+^^ 

22.  ^V^  +  ^P^-y^» 

23.  2  0^(2/  4-2)2  +  2  y2(2  4-  :?:)2  4.  2  z\x  4-  3/)2 

4- 4:^5/3(2:4-^4- 2). 

24.  (a4-^)(«4-2  5)(a4-3  5)(a4-4^>)  +  M. 

25.  x-\-x^  -\-x^-\-2x^  +  2xi  -^2  x^^^, 

26.  «-2  4.  6-4  4-  c2  4-  2  a-15-2  -  2  a-i(?  -  2  6-2c. 

27.  Show  that  a^  ^  53  _^  ^  _  3  ^5^  jg  ^j^g  square  root  of 
(^2  _  J^)3  +  (52  _  cay  4-(<?2  -  a6)3  -  3(a2  -  ^>c)(62  _  ^a) 

28.  Show  that  l-\-(x-l)x(x+l)(x  +  2}  is  a  perfect 
square.    Hence,  write  down  the  square  root  of  1 4-  2  •  3  •  4  •  5. 

29.  If  2:4  H-  4  -  4  ^(^,2  ^  2)  4-  8  2:2  =  (^2  _|_  ^^^  4.  2)2,  what 
is  the  value  of  w? 

30.  Find  the  square  root  of  (-^  -  -^V  ^^  +  1. 


INVOLUTION  AND  EVOLUTION  307 

31.  Prove  that  x^ -\- (x -h  ly  +  (x -\- 2y -\- {x -\- Sy  -  5  is 
a  perfect  square.  If  the  sum  of  the  squares  of  any  four 
consecutive  integers  is  diminished  by  5,  what  may  be 
said  of  the  form  of  the  result? 

236.   Square  root  of  arithmetical  numbers.     We  know 

that  +  VT  =  1,  +  VTOO  =  10,  +  VTOOOO  =  100,  +  VlOOOOOO 
=  1000.  Hence,  the  positive  square  root  of  any  number 
between  1  and  100  is  between  1  and  10;  that  of  any 
number  between  100  and  10,000  is  between  10  and  100, 
and  so  on.  That  is,  the  integral  part  of  a  square  root  of 
a  number  of  two  figures  contains  one  figure;  that  of  a 
number  of  three  or  four  figures  contains  two  figures,  and 
so  on.  Hence,  to  find  the  number  of  figures  in  the  integral 
part  of  the  square  root  of  a  given  number  begin  at  the 
units'  figure  and  separate  the  number  into  periods,  or  groups^ 
of  two  figures  each,  the  last  period  containing  either  one 
or  two  figures  according  as  the  given  number  contains  an 
odd  or  an  even  number  of  figures  in  the  integral  part. 
There  are  as  many  figures  in  the  integral  part  of  the 
square  root  of  the  given  number  as  there  are  periods. 

ILLUSTRATIVE   EXAMPLES 

1.    Find  the  positive  square  root  of  9604. 

Solution.  Since  there  are  two  periods  in  the  given  number,  there 
are  two  figures  in  the  integral  part  of  the  root,  which  we  find  in  the 
same  manner  as  we  did  the  square  root  of  a^  +  2  ab  -\-  h\  For  a,  we 
take  the  greatest  number  of  tens  whose  square  is  less  than  9604  ;  that 
is,  9  tens,  or  90.  In  practice,  we  take  the  greatest  number  whose 
square  is  equal  to  or  less  than  96  in  the  first  period  at  the  left.  The 
work  is  completed  as  follows  : 

96^04190 +  8 
8100 
2  a  =  180  15  04 
2a  + 6  =  180  + 8  =  188  15  04 


308 


ELEMENTARY  ALGEBRA 


Remark. 


In  practice  the  work  is  usually  arranged  thus : 
96'04[98 


81 


188 


15  04 
15  04 


2.    Find  the  positive  square  root  of  56026.89. 

Solution.     Observe   that  if   the   square  root  of   a  number  has 

decimal  places,  the  number  itself  has  double  that  number  of  decimal 

places.     For  this  reason,  in  extracting  the  square  root,  decimals  are 

separated  into  periods  of  two  figures  each  beginning  at  the  decimal 

point. 

5^60^26.89 1 236.7 


-  4 
2a  =  2(20)=  40  160 
2  a  +  6  =  43  1  29 

2a  =  2(230)=  46 

2  a  +  &  =  46 

2a  =  2(2360)=  47 

2  a  +  &  =  47 

0  3126 
6  27  96 

20 
27 

3  30  89 
3  30  89 

237.  Approximate  square  root.  When  a  number  is  not 
a  perfect  square,  we  annex  as  many  periods  of  zeros  as  are 
desired  and  continue  the  process  of  extracting  the  square 
root.  In  this  way  we  obtain  a  rational  number  which  is 
an  approximate  value  of  the  square  root  of  the  number. 

ILLUSTRATIVE  EXAMPLE 

Find  to  two  places  of  decimals  the  square  root  of  10. 
10.00^00^0013.162 


9 


61 


[Too 

61 


626 


39  00 
37  56 


6322 


144  00 
126  44 


INVOLUTION  AND  EVOLUTION  309 

Therefore,  vTO  =  3.16,  correct  to  two  places  of  decimals. 

Remark.     Had  the  third  decimal  figure  been  either  5  or  greater 
than  5,  we  should  have  taken  3.17  as  the  approximate  square  root. 

EXERCISE   113 

Find  the  positive  square  root  of  : 

1.    6724.                     2.    14884.  3.  5776. 

4.    53361.                   5.    110889.  6.  99856. 

7.    591361.                .8.    1723969.  9.  146.41. 

10.    91083.24.           11.    100.2001.  12.  493:817284. 

Find,  to  two  places  of  decimals,  the  square  root  of: 
13.     2.  14.     3.  15.     5.  16.     7. 

17.    11.  18.     13.  19.    12.  20.    15. 

Find,  to  three  places  of  decimals,  the  value  of: 

21.    (l+V2)(l-f-V3).  22.    (7V5-2)(V2-f-l). 

23.    (2V3-hV5)(l  +  3V2).     24. 


V3-V2 


25.    — 26. 


V2-1  V3-4 

Find,  to  two  places  of  decimals,  approximate  values  of 
the  roots  of  the  following  simple  equations: 

27.  (a:-l)V2  =  V3  4-2. 

28.  a:(lH-V2)  =  2a:4-3V3. 

29.  V2a:-l  =  V5  +  2.  30.    a.^  V5-l^ 

2 
Suggestion.    2x-l  =  9+4V5. 


310  ELEMENTARY  ALGEBRA 

238.  Equations  solved  by  finding  the  square  roots  of  a 
number.  Any  equation  which  can  be  reduced  to  the 
form  ax^  =  5  in  which  a  and  h  are  positive  numbers  and 

-  is  a  perfect   square,  has  rational    roots  which  may  be 
a  ^ 

found  by  taking  the  square  root  of  -. 

ILLUSTRATIVE  EXAMPLE 

Solve  the  equation  4a::2  _  9  _  q. 

Solution.  4  a:2  -  9  =  0.  (1) 

Solving  (1)  for  x^,  x^  =  |.  (2) 

Now  a  root  of  (2)  is  a  number  whose  square  is  ^ ;  therefore,  it  is 

a  square  root  of  f .     Hence,  taking  the  square  root  of  each  member 

of  (2), 

x  =  ±|.  (3) 

Therefore,  the  roots  of  (2)  are  +  |-  and  -  |. 

Note.  Any  root  which  can  be  found  by  the  above  method  can 
also  be  found  by  factoring,  as  in  section  117. 

Thus,                                    4  x2  -  9  =  0.  (1) 

Factoring,             (2  a;  -  3)  (2  x  +  3)  =  0.  (2) 

Equating  (2  a;  -  3)  to  0,       2  a:  -  3  =  0.  (3) 

Equating  (2  x  +  3)  to  0,       2  x  +  3  =  0.  (4) 

Solving  (3)  and  (4)  for  x,              ^  =  f  >  ^^d  -  \. 

EXERCISE  114 

Solve  the  equations  in  examples  1-14. 

1.  2^2  =  4.                                  2.  :?^  =  9. 

3.  2  2^2  -  32  =  0.                     4.  2^2  =  a\ 

5.  a2^=^>2.                              6.  182^-200  =  0. 

7.  4  2;2=  49.                            8.  aW^=^\, 

9.  (a  +  6)22:2=a2_2a6  +  62.  10.  {x  -f  1)2  =  16. 

11.  (aa;  H- 6)2  =  c2.                    12.  a:2_f_2a:+l  =^2^2  a  +  1. 


INVOLUTION  AND  EVOLUTION  311 

13.    (2a: -3)2  =  (a -2)2.  14.    f^±lj  =  l. 

15.  In  the  illustrative  example,  page  310,  in  taking  the 
square  root  of  each  member  of  (2),  why  is  not  equation 
(3)  written  ±  a:  =  ±  |  ? 

Calculate,  to  two  places  of  decimals,  the  numbers  which 
satisfy  the  following  equations  : 

16.  a;2  =  2.  17.    x^  =  S.  18.    0^2  =  5. 

19.    2:2  =  32.  20.    2^=27.  21.    2:2  =  32.16. 

22.  Solve  the  equation  ^22:2  _|_  2  abx  -\-  b^  =  c^. 

23.  Solve  the  equation  4  pV  —  4  pqx  =  p^  —  (f, 

4 

24.  Solve  the  equation  x=  -- 

X 

9 

25.  Solve  the  equation  2:  —  2  = 


2:-2 

26.  For  what  values  of  x  will  the  H.  C.  F.  of  2:^  __  3  jp2 

4-  2:  —  3  and  2:^4.2  2^2 +  2^  +  2  reduce  to  5? 

3  2:  —  2  7 

27.  Solve  the  equation  — - —  = • 

(  o  X  —  2t 

28.  If  2:  =  2  satisfies  the  equation 

(a:  _  3)(2:  +  1)  =  3  2:(2:  4- 2)  -  (2:  +  a2), 
what  are  the  values  of  «  ? 

2'  2        X    I     1 

29.  Solve  the  equation  -=         ^. 

^  2;  +  4      2;-2 

30.  When  2  2:*  4-  10  2:3  +  13  2:2  -  2:  -  6  ig  divided  by 
2:2  4-  3  2:  +  2,  the  quotient  is  2  2:2  -|-  m22:  —  3.  Find  the 
values  of  m, 

31.  If  w2  -  1  =  2:(2:  4-  X){x  4-  2)(2:  +  3),  find  the  values 
of  m  in  terms  of  x. 


CHAPTER  XII 

QUADRATIC   EQUATIONS 

Definition.  An  equation  whose  second  member 
has  been  reduced  to  zero  by  transposition  of  terms  is  called 
an  equation  of  the  second  degree,  or  simply  a  quadratic 
equation  in  one  unknown  number,  as  x^  when  its  first 
member  is  a  polynomial  of  the  second  degree  in  x. 

Thus,  5  +  2a;2-7x  =  0,  and  m  -  nx^  -\-  qx  ■\-  n^  -  p^  =  0  are 
quadratic  equations. 

240.  Notation.  It  is  customary  to  arrange  the  poly- 
nomial in  the  first  member  of  a  quadratic  equation  accord- 
ing to  the  descending  powers  of  x  and  to  render  positive, 
when  necessary,  the  coefficient  of  the  highest  power  of  x 
by  multiplying  both  members  of  the  equation  by  —  1. 
The  first  member  of  a  quadratic  equation  being  a  tri- 
nomial of  the  second  degree  in  rr,  has,  in  general,  three 
terms,  a  term  in  x^  with  positive  coefficient,  a  second  term 
in  X,  and  a  third  term  which  does  not  contain  x  and  which 
is  the  constant  term  in  the  equation.  In  any  particular 
equation,  however,  not  all  of  these  terms  may  occur,  as  the 
coefficient  of  one  or  more  of  them  may  reduce  to  zero  ; 
but  we  shall  assume  that  in  a  quadratic  equation  the 
coefficient  of  x^  is  not  zero. 

241.  Standard  form  of  a  quadratic  equation.  Any  quad- 
ratic equation  can  be  reduced  to  the  standard  for m^ 

ax^  -{-bx-^  c  =  0. 

In  this  standard  form  a  is  positive  and  different  from 
zero,  while  h  and  c  may  have  any  values,  including  zero, 

312 


QUADRATIC  EQUATIONS  313 

Thus,  the  equation  mx^  —  3  mz  +  5  =  nx^  —  2  n  +  3  /)x  —  4,  when 
transformed,  becomes  {m  —  n)a:2  —  3(7n  +  i»)2:  +  (2  n  +  9)  =  0.  In  this 
last  equation  the  a,  6,  and  c  of  the  standard  form  have  the  following 
values : 

a  —  m  —  n^   h  =  —  3(m  +  jo),    and    c  =  2  n  +  9. 

242.  Complete  quadratic  equation.  A  quadratic  equa- 
tion in  which  none  of  the  coefficients  a,  6,  or  e  reduces  to 
zero  is  called  a  complete  quadratic  equation.  All  other 
quadratic  equations  are  incomplete  quadratic  equations. 

243.  Solution  of  incomplete  quadratic  equations.      The 

roots  of  an  incomplete  quadratic  equation  can,  jn  general, 
be  obtained  by  methods  of  solution  which  have  been 
treated  in  preceding  chapters.  These  methods  are 
applied  in  the  illustrative  examples  which  follow. 

•  ILLUSTRATIVE  EXAMPLES 

1.  Solve  the  equation  2x^  =  0. 

Solution.  2  x^  =  0  is  an  example  of  a  quadratic  equation  in 
which  two  of  the  coefficients,  namely  h  and  c,  reduce  to  zero.  It  is 
evident  that  the  only  value  of  x  which  satisfies  this  equation  is  0, 
since  from  2x^  =  0,  we  derive  x^  =  0.  We  say  that  the  equation  has 
two  roots  each  equal  to  0,  one  root  corresponding  to  each  factor  of  x^- 

2.  Solve  the  equation  2a;2  —  3  a;  =  0. 

Solution.  2  a:^  —  3  a;  =  0  is  an  example  of  a  quadratic  equation 
in  which  the  constant  term  c  reduces  to  zero.  We  solve  such  an 
equation  by  the  method  of  section  117.     Thus,  factoring, 

2x^-%x  =  x(2x-  3)=0.  (1) 
Equating  each  factor  of  (1)  to  zero,  we  have  (2)  and  (3), 

a:  =  0.  (2) 

2a:- 3  =  0.  (3) 

Solving  (3)  ar  =  |.  (4) 

Therefore,  the  roots  of  2  a:^  —  3  a:  =  0  are  0  and  4,  which  may  be 
verified  by  substituting  each  of  these  results  in  (1). 


314  ELEMENTARY  ALGEBRA 

Remark.  Observe  that  when  the  constant  term  c  in  a  quadratic 
equation  is  zero,  one  root  of  the  equation  is  zero,  and  conversely, 
when  one  root  of  a  quadratic  equation  is  zero,  the  constant  term  is 
zero. 

3.  Solve  the  equation  Sa;^  —  2  =  0. 

Solution.  3  a:^  —  2  =  0  is  an  example  of  an  equation  in  which  i, 
the  coefficient  of  x,  is  zero.  Such  an  equation  is  sometimes  called  a 
pure  quadratic  equation. 

From  3  a;2  —  2  =  0,  we  derive  x^  =  |^. 

The  values  of  x  which  satisfy  this  equation  are  evidently  the  square 

2  /2  "v/fi 

roots  of  -.     Hence,  x  =  i'V- ,  or  ±  -7—  • 
o  '33 

Therefore,  the  roots  of  Sa:^  —  2  =  0  are    —   and ,   which 

3  3 

may  be  verified  by  substituting  these  numbers  in  the  given  equation. 

4.  Solve  the  equation  a^^  _|_  4  =  0. 

Solution.  a;2  +  4  =  0  is  an  example  of  a  pure  quadratic  equation, 
both  of  whose  terms  have  the  same  sign.  No  positive  or  negative 
number  exists  which  satisfies  this  equation;  for  the  sum  of  two 
positive  numbers,  one  of  which  is  not  zero,  cannot  be  zero.  This 
type  of  quadratic  equation  will  be  considered  in  section  251. 

EXERCISE  115 
Solve  the  following  incomplete  quadratic  equations : 

1.  5a;2  =  0.  2.  x^-2x=(i. 

3.   3a;2-.4a:  =  0.  4.  4a^2_i,=  o. 

5.  5a;2-3a:  =  0.  6.  7a:2_15  =  0. 

7.  5a:2~2  =  0.  8.  2a:2-5:r  =  0. 

9.  8:c2_9  =  o.  10.  ax^-hx  =  0. 

11.  9a;2~16a:  =  0.  12.  3a:2-5  =  0. 

13.   Qx^-bx  =  0.  14.   20a:2-8a:=0. 

15.  2:^2-11=0.  16.  aV-h^^a. 

17.   (a+h)x^-(^c-\-d)x=0.   18.   (a-h6)a;2-(a~6)2:=0. 


QUADRATIC  EQUATIONS  315 

19.  5ax^-Sb  =  0.  20.    ^^^:=0. 

Z  b 

21.   ax^—a  =  0.  22.  x^  —  5  =  0, 

23.    ^-x  =  0.  24.  --8a:  =  0. 

X  X 

25.    ^x^  +  ^x=0.  26.    19:c2_i2:  =  0. 

244.  "Completing  the  square."  From  section  92  we 
know  that  the  trinomials  x^ -\- 2  ax -\-  a^  and  x^  —  2  ax-i-  a^ 
are  both  perfect  squares.  We  therefore  infer  that  both 
x^-\-2  ax  and  x^  —2  ax  are  converted  into  perfect  squares 
by  the  addition  of  a^  to  each.  Observe  in  each  case  that 
a^  may  be  obtained  by  taking  the  square  of  one  half 
the  coefficient  of  x.  We  therefore  have  the  following 
rule  for  completing  the  square  of  a  binomial  of  the  form 
x^  -\-2  ax,  in  which  the  coefficient  of  a;^  is  1 : 

Rule.     Add  the  square  of  one  half  the  coefficient  of  x. 

EXERCISE  116 

Complete  the  square  in  each  of  the  following  examples, 
and  state  the  binomial  whose  square  is  obtained  : 

1.     X'^+2X.  2.     X^-2X.  3.     X^-\-4:X. 

4.  x^-^-Qx.  5.  x^-6x,  6.  x^-\-16x. 

7.  x^  -  18  a;.  .  8.  x^  +  20  a;.  9.  x^  +  ^x. 

10.  x^  —  ^x.  11.  x^  —  ^x.  12.  x^-{-^x. 

13.  x^-\-2cx.  14.  x^  -  2(^a -\- b}x.  15.  a;2  +  2(a-6>. 

16.  x^  —  ax,  17.  x^-\-Sx.  18.  2;2  —  (a  4-  h}x. 

19.    x^-^x,         20.   x^  +  ^x,  21.   x^-^^'^x, 

0  bd  c?+  d 


316  ELEMENTARY  ALGEBRA 

245.  Solution  of  a  quadratic  equation  by  ^^  completing 
the  square."  When  the  roots  of  a  quadratic  equation  are 
not  rational  numbers,  its  solution  by  factoring  [section 
117]  is  not,  in  general,  so  convenient  as  that  explained 
in  the  following  illustrative  examples  : 

ILLUSTRATIVE  EXAMPLES 

1.    Solve  the  equation  Sx^  —  2x  —  2  —  0. 

Solution.  3  x2  -  2  a;  -  2  =  0.  (1) 

Dividing  both  members  of  (1)  by  3  so  that  the  coefficient  of  x^ 

shall  be  1,  x2-|a;-|  =  0.  (2) 

Transposing,  x^  —  ^x  =  ^.  (3) 

Completing  the  square  by  adding  the  square  of  one  half  the  coeffi- 
cient of  X  to  each  member  of  (3), 

^^-t^  +  i  =  l  +  i  =  f  W 

Since  the  first  member  of  (4)  is  a  perfect  square,  we  have, 
Extracting  the  square  root, 


Transposing  and  combining, 

''-      3      • 


(7) 


Therefore,  the  roots  of  3  a;^  -  2  a:  -  2  =  0  are  ^  \       and  ^         '^ . 

3  o 

2.    Solve  the  equation  4:p^x^  —  4  mpx  -{-m^  —  m  —  n  =  0. 
Solution.    4:p^x^  —  4  mpx  +  m*  -  m  -  n  =  0.  (1) 

Dividing  both  members  of  (1)  by  4:p^, 

^2_m^^  m^-m_-n^O  (2) 


p  4jo 


m  -i-  n  —  m' 


Transposing  in  (2),  x^-'^x  =  '"  -^  '  ~  ""  .  (8) 

p  4/>^ 


QUADRATIC  EQUATIONS  317 

Completing  the  square  and  combining  in  (3), 

p^4:p^~   4j9«   •  ^*^ 

Since  the  first  member  of  (4)  is  a  perfect  square,  we  have, 


V 

Extracting  the  square  root, 

2pJ         ip' 

• 

Transposing, 

2p         '                  ^^ 

Therefore,  the  two  roots  of 

the  riven  eouation  are  ^  +  ^^  +  » 

2p 

2p 

Remark.  The  solution,  example  2,  illustrates  the  solution  of  a 
literal  quadratic  equation  by  completing  the  square.  Had  the  object 
been  merely  to  solve  the  equation,  the  solution  would  be  as  follows : 

Transposing,  i p^x^  —  4:  mpx -\- m'^  =  m  ■}- n.  (2) 

Since  the  first  member  of  (2)  is  a  perfect  square, 

(2px  -  m)2=  m  +  n.  (3) 


Whence,  2px  —  m  =±  Vm  +  n 


and  X  =  ^'±^^  +  ^, 

2p 

EXERCISE  117 

Solve  the  following  quadratic  equations  by  the  method 
of  completing  the  square  : 

1.    rr2  -{-  4 re  -f  3  =  0.  2.    x^  +  4:X-hl  =  0. 

3.    x^-{-2x-i  =  0.  4.    a:2  H-  3  2^  +  1  =  0. 

5.    x^-\-5x-\-b  =  0.  6.    2;2  +  10.r4-15  =  0. 


318  ELEMENTARY  ALGEBRA 


7. 

2:2  +  112:4-25  =  0. 

8. 

2:2-3a:-l  =  0. 

9. 

a;2  -  5  a:  +  3  =  0. 

10. 

2:2-7x4-11  =  0. 

11. 

a;2_lla;-l  =  0. 

12. 

a:2  _  13  a:  +  30  =  0. 

13. 

x^-15x-5  =  0. 

14. 

2:2-102:4-23  =  0. 

15. 

x^-6x+4:  =  0. 

16. 

4a:2_42:-l  =  0. 

17. 

Sx^-]-Sx-2  =  0, 

18. 

52:2-52:4-1  =  0. 

19. 

Sx^-lx+S  =  0. 

20. 

72;2_7a;-5  =  0. 

21. 

llx^-hlx-S  =  0. 

22. 

52:2-32:-5  =  0. 

• 
23. 

13  2^2  -  13  2;  -  3  =  0. 

24. 

5a:2_52:-l  =  0. 

25. 

2x^-Sx-i=::0. 

26. 

(a;_5)(2:-3)=l. 

27. 

(x-6)(x-S)  =  4. 

28. 

2:2- (a  4- 1)2: 4- a  =  0. 

29.  52:2  -  «(5  4- 1)2:  4- «2  =  0. 

30.  (a  4- 3)2:2 -2(a  4- 4)2;  4- (a  4- 5)  =0. 

31.  («4-^')2:2  4-(5  4-2)2:-(a-2)=0. 

32.  (a  -  2)2:2  +  ^  __  («  _  3)  =  Q. 

33.  (2  a-  b)x^  -Sax-\-(a-\-b)=0. 

34.  2:2  4-2(w4-l)a:4-m(w  +  l)=0. 

35.  4  2:2  4-  4^0^  ^  ^^^  4-  m2  4-  ri2  _  Q, 

36.  2;2  -  2  aa;  4-  ^2  -  a  =  0. 

37.  42:2- 4^2:4- 52_  4^  =  0. 

38.  (a  4-  5)22:2  -  2(a  +  6)22:  +  (a  4-  5)2  -  2  =  0. 

39.  16  a^V  -  8  ab(a  +  5)2: 4-  («  4-  5)2  -  16  a^^  =  0. 

40.  («  4- 5)22:2  4- 2(^2 +  52)2:  + (a -5)2=0. 

In  the  following  examples,  clear  the  equation  of  frac- 
tions and  solve  the  resulting  integral  equation,  checking 
the  roots  found : 

12         4 

41.  -4. 


X     x  —  1      3 


45. 


53. 


QUADRATIC   EQUATIONS  319 


42.  -^ §-+J-  =  0. 

2:4-1       2:  +  212 

43.     ^ +  1+ =  0. 

32^-3         ^2x-S 


3 


+  1  -  TT^-T^  =  0. 


22:  +  l  3a;  +  2 

12  5 


62:  —  5a      a      a  —  6x 


*  a;-2       a;+2  *  *    ix-\-2      3a:  +  5 

48     -1^+^^+1  =  1  49      9^+1    ,  62^+5^8 

•  6x-5"^4a:H-3  '    15x  +  l      3a:  +  5      9* 

50.    ^±i 2;-2  a;-2       ^^ 

2  2:2+32:-22  2:2  +  2;-1^2;2  +  3a:H-2 

51.    ^±^ +  _^±8_     ^-4 ^ 

2  2:2  +  52:  +  22;2_2._6^.2  2:2-52:-3 

2a:  +  2       ,  4:x-{-2         ^      3a:  +  3 

rv n "T" 


3a:2  4.a;-2      6x^-Ux+6      2x^-x-^ 

3  ^  =  ^  . 

x^  +  Sx-\-2     x^-\-7x  +  12     x^-^4x-\-S' 


246.  Approximations.  Many  of  the  problems  which 
occur  in  physics  and  geometry  give  rise  to  quadratic 
equations.  In  general,  the  roots  of  such  quadratics  are 
irrational  numbers  which  appear  in  the  form  of  radical 
expressions.  For  practical  purposes  we  usually  require 
rational  results  which  give  approximately  the  values  of 
the  roots.  The  method  used  in  obtaining  such  approxi- 
mations may  be  seen  from  the  following : 


320  ELEMENTARY  ALGEBRA 

ILLUSTRATIVE  EXAMPLE 

Approximate  to  two  decimal  places  the  roots  of 

25^  +  2     j_47^jf31^^ 
x+1  2x-S 

Solution.     25X  +  2  47.-f31^Q 

x+1  2x-d 

Clearing  (1)  of  fractions  and  combining, 


(1) 


a;2_30a;_  8  =  0.  (2) 

Solving  (2),  a:  =  15  db  V233.  (3) 

That  is,  x  =  15±  15.264+  (4) 

Therefore,  the  roots  of  the  given  equation,  correct  to  two  places 
of  decimals,  are  30.26  and  —  .26. 

Remark.  —  In  checking  the  roots  of  a  quadratic  equation  whose 
roots  are  irrational,  the  expressions  in  terms  of  radicals  should  be 
substituted  for  the  unknown  in  the  given  equation.  Thus,  in  check- 
ing the  result  in  the  foregoing  example,  substitute  15  ±  V233  for  x 
in  the  given  equation.  The  rational  numbers  obtained  as  approxi- 
mations will  not  satisfy  the  equation. 


EXERCISE  118 

Approximate  to  two  places  of  decimals  the  roots  of  the 
following  equations : 

1.    a^_4a;  +  2  =  0.  2.  a^-6x-{-l  =  0. 

3.    a^-22x-hllS  =  0.  4.  2:2_  20a: +  95  =  0. 

5.    4a;2_i2a:-3  =  0.  6.  2x^-\-Sx-6==0. 

7.    3a:2-5a:-l  =  0.  8.  5x^-1  x +  1  =  0. 

9.    l2^-15x-\-5  =  0.  10.  a:2_32a;-l  =  0. 

U.    r^-102:  +  8  =  0.  12.  3a:2_8a,4.i=:0. 

13.    2x^-lSx-\-l  =  0.  14.  3a:2  +  5a:-3  =  0. 

15.    ^a^  +  lx-2  =  0.  16.  8a^2- 282:4-21  =  0. 


QUADRATIC  EQUATIONS  321 

247.  Irrational  equations.  The  unknown  number  in 
an  equation  sometimes  occurs  in  expressions  which  are 
found  under  radical  signs.  Such  equations  are  called 
irrational  equations. 

Thus,  Qx  —  y/Wx  +  4  =  20  is  an  irrational  equation. 

We  shall  consider  only  equations  in  which  the  square 
root  of  expressions  containing  the  unknown  number  is 
indicated.  The  solutions  of  the  following  examples  will 
illustrate  the  method  of  solving  such  equations. 

ILLUSTRATIVE  EXAMPLES 


1.    Solve  the  equation  bx  —  V3  a:  +  7  =  11. 


The  equation  5  jl-  -  V3  x  +  7  =  11  is  an  example  of  an  equation  in 
which  only  a  single  square  root  occurs. 


Solution.  5  a;  -  V3  a:  +  7  =  11.  (1) 

Transposing  terms  so  that  the  radical  expression  stands  alone  in 
one  member  of  the  equation, 


-  v/3a:+  7  =  -  5a:  +  11.  (2) 

Squaring  both  members  of  (2),  3  x  +  7  =  25  x^  -  110  x  +  121.  (3) 

Simplifying  (3),       25  x*  -  113  x  +  114  =  0.  (4) 

Factoring,                   {x  -  3)  (25  a;  -  38)  =  0.  (5) 
Therefore,  the  roots  of  equation  (4)  are  3  and  |^. 
Substituting  3  in  the  given  equation, 

15_V9TT:=11.  (6) 
Simplifying  (6),  11  =  11,  which  is  an  identity. 
Substituting  ^  in  the  given  equation. 


5x  ||-V3xff +  7  =  11.  (7) 

Simplifying  (7),  %!  =  11, 

which  is  false.  Since  ||^  does  not  satisfy  the  given  equation,  it  is  no^ 
a  root  of  the  equation.  It  may  readily  be  verified  that  ||^  is  a  solu 
tion  of  the  equation  5  x  +  V3  x  +  7  =  11,  which  equation  differs  from 
the  given  equation  only  in  the  sign  of  the  radical  expression. 


322  ELEMENTARY  ALGEBRA 

Note.  In  the  process  of  squaring,  as  in  the  above  solution,  we 
multiply  each  member  of  the  equation  by  a  factor  containing  the 
unknown  number.  The  resulting  rational  equation  may  have  solu- 
tions which  are  not  solutions  of  the  given  equation.  It  is,  therefore, 
necessary  to  test  each  root  of  the  rational  equation  by  substituting  it 
in  the  given  equation. 


2.    Solve  the  equation  Vrr  +  5  —  V7  a;  +  4  -f  V2  rc4-  9  =  0. 

(The  first  step  in  the  solution  of  an  irrational  equation  of  this 
form  is  to  arrange  the  terms  so  that  one  radical  shall  stand  alone  in 
one  member.  The  process  of  squaring  leads  to  an  equation  in  which 
a  single  radical  occurs  and  the  solution  proceeds  as  in  that  of 
example  1.) 


Solution.     Va;  +  5-V7a;  +  4  +  V2a:-f  9  =  0.  (1) 

Transposing,  Va;  +  5  +  V2z  +  9  =  V7 x  -\- 4.  (2) 

Squaring  (2)  and  simplifying,        2  a:  —  5  =  y/(2x-\-9)(x-\-5).     (3) 
Squaring  (3)  and  simplifying, 

2x2 -39x- 20  =  0.  (4) 

Factoring,  .  (2  x  +  1)  (a:  -  20)  =  0.  (5) 

Therefore,  the  roots  of  equation  (4)  are  20  and  —  i. 
Substituting  20  in  the  given  equation, 

V25  -  Vl44  +  V49  =  0.  (6) 

Simplifying  (6),  5  —  12  +  7  =  0,  which  is  an  identity. 

Substituting  -  ^  in  the  given  equation, 

V|-VJ-fV8  =  0.  (7) 

Simplifying  (7),  3V2  =  0, 

which  is  false ;  hence,  —  4  is  not  a  solution  of  the  given  equation. 
It  may  readily  be  verified  that  —  |  is  a  solution  of  the  equation 

-  y/x  +  6  -  \/7a:  +  4  +  V2x  +  9  =  0. 


EXERCISE    119 

Solve  the  equations  : 


1.    V:r-1-1  =  0.  2.    V2a:-8-3  =  0. 

1 


■Vx  +  l 


=  2.  Suggestion.     Clear  of  fractions. 


QUADRATIC  EQUATIONS  323 


1 


4.    V52:H-2  =  7.  5.  -5  =  0. 

V2a:-3 

6.    x-{-Vx  =  12.  7.    a;-2Vi-8  =  0. 


8.    x  +  ^Sx-i-1  =  7.  9.    a:-V3a:+7=lll. 


10.    5x-\-V5x-^4  =  52.  11.    22:-V2a:-f  3  =  17. 


12.    3a:-|-V3a;  +  2=4.  13.    5rr  +  Va;  +  2  =  ^. 

14.    3Vi-Va:  +  16  =  4.  15.    V2a:  +  9  +  7  =  3V2^. 


16.    Va;-7  +  3=V2^ 


iC. 


17.    Va;  +  24-V2;-15  =  3. 


18.    V92:-f  l-V4a;-3  =  3. 


19.  V-35-99a:  +  V27  +  2a:  =  13. 

20.  V7^T4-f2V3^-VT5^T76  =  0. 


21.    V2a:H- ll  +  2Va:  +  2  =  V20a:-19. 


22.    V2:-7-V:c-ll=V3a:-29. 


23.    V2a;-6-Va:-l+V3a;-15  =  0. 


24.    V2a;4-l4-V3a:-ll-V92:-8  =  0. 


25.  V7a;  +  l-V3a:  +  10  =  l. 

26.  a;^  —  12x"^  4- 1  =  0.        Suggestion.     Clear  of  fractions. 

^2a;H-l  ^       2: 

28.  a-\-b  —  Va^  —  a:  =  Vi^  -|-  x. 

29.  wax-{-b  =  — — » 

V22: 

30.  2Vi  =  Va  + V42;— <?. 

31.  Vi  +  Vx  -\-  a  = 


^x-\-a 


324  ELEMENTARY  ALGEBRA 

32.    V^+V^       '' 


X-4         ^X-\-4:  3 


33..    ■\/x  +  2a  —  Wx-\-2b  =  2Vx, 
34.    Vx^-{- 5ax—2a^  =  x-{- a. 


35.     Va:  +  a^  =  «  -h  Va;. 


36.     ^^  +  ^  +  ^^^-^=6. 
^a-\-  x  —  ^a  —  x 

248.  Solution  of  a  quadratic  equation  by  formulae.     Any 

quadratic  equation  can  be  reduced  to  the  standard  form, 

ax^  -\-bx-\-  e=0. 

By  this  is  meant  that  by  assigning  particular  values  to 
the  coefficients  a,  6,  and  o,  this  equation  reduces  identically 
to  any  given  quadratic  equation.  The  solution  of  this 
general  equation,  therefore,  contains  the  solution  of  any 
given  quadratic  equation.  The  solution  of  a  given 
quadratic  equation  may,  therefore,  be  found  by  substitut- 
ing in  the  formulae  which  give  tlie  roots  of  the  general 
quadratic  ax^  -\-  bx-{-  c  —  0,  the  proper  values  of  the  coeffi- 
cients a,  5,  and  c. 

The    derivation    of    the    formulae    for    the    roots    of 
.  ax^  -1-  62; -I-  (?  =  0  is  as  follows  : 

ax^  +  fta:  +  c  =  0.  (1) 

Since  a  is  not  zero,  x^  +  -x  +  -=  0.  (2) 


(3) 


Transposing 

U 

X 

a 

a 

Completing 

the 

square, 

2  .  f> 
a 

^ia^~ 

b'^-iac 
4a3 

(*) 


QUADRATIC  EQUATIONS  325 

Thatis,                      (x  +  -)   =-^^^.  (5) 

Extracting  the  square  root,  

6       ±  V62  -  4  ac  ,«x 


Transposing  and  simplifying, 


2a 


(7) 


Representing  the  two  roots  of  aa^  -\- bx -\-  c  =  0  hy  x^  and 
x^,  we  have,  therefore,  the  following  formulae, 


_   -fr-V&2_4gg 

Note.  When  (6^  _  4  ac)  is  negative,  the  given  equation,  ax^  +  hx 
+  c  =  0,  is  satisfied  by  no  positive  or  negative  number.  The  ex- 
pression \b^  —  4:  ac  is,  for  the  present,  without  meaning  when 
62  _  4  ac  <  0.     [See  section  251.] 

ILLUSTRATIVE   EXAMPLES 

1.    Solve  the  equation  12x^  -{-  x  —  6  =  0. 
Solution.     Here  a  =  12,  6  =  1,  c  =  —  6 ;  hence,  by  substitution  in 
the  foregoing  formulae  we  find  the  roots  to  be 


1  +  VI  +  288  ajj(j  -  1  -  VT+  288 


24  24 

which,  when  simplified,  are  equal  to  ^  and  —  J,  respectively. 

2.    Solve  the  equation  2  mx^  —  (3  w  +  l)a7  +  (m  +  1)  =  0. 
Solution.     Here  a  =  2  m,  6  =  -  (3  m  +  1),  c  =  m  +  1;  hence,  by 
substitution  in  the  foregoing  formulae  we  find  the  roots  to  be 


3  m  +  1  +  V(3  m  4-1)2-8  7/?(m  +  1) 
4m 


and  3  m  +  1  -  V(3  m  +  1)2-  8  m(m  +  1) 

4  m 

which,  when  simplified,  are  equal  to  1  and  ^  "^     ,  respectively. 

2  w 


326  ELEMENTARY  ALGEBRA 

EXERCISE  120 

Find  the  roots  of  the  following  equations  by  substitut- 
ing, in  each  case,  the  proper  values  of  a,  6,  and  c  in  the 
foregoing  formulae. 

1.  a:2-5a;-14  =  0.  2.  x^-10x  +  21  =  0, 

3.  6ir2-a^-2  =  0.  4.  x^-%x-2^  =  0. 

5.  20a;2-23ar  +  6  =  0.  6.  15  a;2- 11  a;- 12  =  0. 

7.  2:2  +  42^  +  2  =  0.  8.  a;2-6a;  +  6  =  0. 

9.   2^2-10  2: +  18  =  0.  10.   a;2-3  2:+l  =  0. 

11,  3  2^2  +  4  2^-1  =  0.  12.  25  2^-10  2:- 2  =  0. 

13.  2^2  —  (c  —  d)x  —  cd  =  0. 

14.  2^2  —  (2  m  -  3  71)2;  —  6mw  =  0. 

15.  abx^-(a^-\-b^)x  +  ah=0, 

16.  4chy^-{-a^=b^-\-4:acx, 


17         1      + 

1 

_-l  +  l. 

2;—  a 

2:  — 

b      a      b 

18.   6c2:2  +  2 

m2: 

+  a^>  =  0. 

19. 

a;- 100  =  10 -Vi. 

20.    2(2:-l)=V2^. 

21. 

a;  +  Vi  =  0. 

22.   2:2-9  =  0. 

23. 

2:2  _  4  =  0. 

24.   2:2-3  =  0. 

25. 

2:2- 3  2:=  0. 

26.   2:2_a;V2  =  0. 

27. 

2:2  +  a:V3  =  0. 

28.    2:2- a  =  0. 

29. 

W2:2  —  n=0. 

30.    «2^2_52  3,0. 

249.   Relations  between  roots  and  coefficients.     Taking 
the  standard  form  of  the  quadratic  equation,  we  have  : 

ax^  -\- hx  -\-  c  =  0.  (1) 


QUADRATIC  EQUATIONS  327 

Dividing  both  members  of  (1)  by  a  (since  a  is  not  zero), 

(2) 


(3) 
(4) 


a        a 

=  0. 

From  the  formulae  of  section  248  we  ] 

tiave  the  roots  of 

^  _  -  6  +  V62  - 
^              •     2a 

4ac 

2a 

■4ac 

From  (3)  and  (4)  by  addition, 

-    ^■— '^^ 

a 

From  (3)  and  (4)  by  multiplication, 

(5) 


(-b  +  y/b^-4:  ac)(-  b  -  V6''  -4  ac)  ' 
4  a2 

4  a** 
=  _l-[6._*.  +  4ae]=£.  (6) 

We  see  from  identity  (5)  that  the  sum  of  the  roots  of 
equation  (2)  differs  in  sign  only  from  the  coefficient  of  x 
in  that  equation.  Also,  we  see  from  identity  (6)  that 
the  product  of  the  roots  of  equation  (2)  is  the  constant 
term  in  that  equation.      Hence, 

In  any  quadratic  equation  of  the  form  ax^  -\-  hx  -\-  c  =  Oi 

(I)  The  coefficient  of  x  with  its  sign  changed  divided  hy  the 
coefficient  of  x^  is  equal  to  the  sum  of  the  roots. 

(II)  The  constant  term  divided  hy  the  coefficient  ofx'^  is  equal 
to  the  product  of  the  roots. 

250.  Formation  of  the  equation.  The  principles  of  sec- 
tion 249  enable  us  to  form  the  quadratic  equation  when 
its  roots  are  given  numbers.     For  this  purpose  we  suppose 


328  ELEMENTARY  ALGEBRA 

the  equation  written  in  the  form  x^+px-hq  =  0^  in  which 

p  and  q  are  written  instead  of  -  and   -,  respectively. 

a  a 

From  identities  (5)  and  (6),  section  249,  we  have 
■^i  +  -^2  =  -^=-A      . 

ILLUSTRATIVE  EXAMPLES 

1.    Form  the  equation  whose  roots  are  3  and  —  2. 

Solution.     Here         x^  +  x^^^  —  2  =  1=  —  p; 
also,  XjXg  =  3x(—  2)  =  — 6  =  ^. 

Hence,  jt>  =  —  1  and  ^  =  —  6. 

Substituting  these  values  of  p  and  ^  in  x^  -{■  px  -{■  q  =  0,  the  re- 
quired equation  is     a;^  —  x  —  6  =  0. 

X  -{•  1  _        X 

5     ~6-6a; 
2 
is  - ;  find,  without  solving  the  equation,  the  other  root, 
o 

Solution.  Reducing  the  given  equation  to  the  standard  form,  we 
>*^^'  6x2+ 5a; -6  =  0.  (1) 

From  (II),  section  249,  the  product  of  the  roots  of  equation  (1)  is 
equal  to  the  constant  term  divided  by  the  coefficient  of  x^ ;  namely, 

\     -6  1 

to  ,  or  -  1. 

6 

Since  |  is  known  to  be  a  root,  the  other  root  is  equal  to  —  1  -^  J ; 
that  is,  the  second  root  of  the  given  equation  is  —  |. 

We  may  check  thi^  result  by  making  use  of  (I),  section  249,  from 
which  we  know  that  the  sum  of  the  roots  of  this  equation  is  the 
coefficient  of  x  with  the  sign  changed  divided  by  the  coefficient  of  x\ 

which  is   —5^6,  or  ^— .     Since  %  is  one  root,  the  other  root  is 
6  3 

equal  to  -^-|=-|  =  -|^,  which  is  in  agreement  with  the  pre- 
ceding result. 


2.    Given  that  one  root  of  the  equation 


QUADRATIC  EQUATIONS  329 

EXERCISE  121 

1.  State  at  sight  the  sum  and  the  product  of  the  roots 
of  the  following  six  equations.  Do  not  solve  the 
equations. 

a:2_22;-|-l=0.  x^-lx-^12  =  0. 

ax^  ~  bx-\-  c=0.  mx^  -^nx-\-pq  =  0. 

2.  One  root  of  each  of  the  following  six  equations  is 
—  2.  Find  the  second  root  in  each  case.  Do  not  solve 
the  equations. 

a:2  +  3a:+2  =  0.  x^-x-Q  =  0. 

3  2;2-h42:-4  =  0.  x^-\-4x-\-4  =  0. 

ax^  +  (2a-\-b)x  +  2b  =  0.        px^ +  (2p  -  q}x- 2  q=  0. 

3.  In  a  pure  quadratic  equation  what  is  the  sum  of  the 

roots  ? 

Form  the  quadratic  equations  which  have  the  following 
roots : 

4.  2  and  —  3.  5.    3  and  2.  6.    —  1  and  3. 
7.    6  and  -  3.                            8.-2  and  -  3. 

9.    1+V2andl-V2.  lo.    3  -  V2  and  3 -|- V2. 

11.    m-\-n  and  m  —  n.  12.    -  +  -  and  2. 

b      a 

13.    m  -f-  Vw  and  m  —  Vw.         14.    f  and  —  |. 
15.    f  V2  and  -  f  V2. 

251.^  Quadratic  equations  in  which  bf^  —  ^ac  is  negative. 
The  equation  a;2  +  4  =  0  [section  243,  example  4],  which 

*This  section  may  be  omitted,  if  so  desired,  until  the  subject  is 
reviewed. 


330  ELEMENTARY  ALGEBRA 

is  of  the  form  aa?  -|-  62:  +  c  =  0,  where  ^  —  4  ac  <  0  [sec- 
tion 248,  note],  is  an  example  of  a  large  class  of  equations 
which  have  no  rational  or  irrational  solution.  There  are 
no  rational  or  irrational  numbers  which  satisfy  an  equa- 
tion such  as  a^  +  4  =  0. 

It  is  evidently  desirable  that  all  equations  should  have 
solutions,  but  this  is  manifestly  impossible  so  long  as  the 
number  system  of  algebra  includes  only  rational  and 
irrational  numbers. 

In  agreement  with  the  generalizing  spirit  of  algebra, 
the  number  system  is  so  extended  that  all  equations  shall 
have  solutions. 

252.  Pure  imaginary  number.  The  square  root  of  a 
negative  number  is  called  a  pure  imaginary  number. 

Thus,  V  —  4,  also  V  —  2,  are  pure  imaginary  numbers. 

253.  Real  numbers.  All  rational  and  irrational  numbers 
are  called  real  numbers. 

Thus,  2,  -  3,  I,  -  I,  Vi,  a/2,  are  real  numbers. 

254.  The  imaginary  unit.  The  pure  imaginary  number 
V—  1  is  called  the  imaginary  unit  and  is  denoted  by  the 
letter  i. 

Thus,  I  =  y/'^\. 

255.  A  pure  imaginary  number  expressed  in  terms  of  i. 

A  pure  imaginary  number  is  expressed  in  terms  of  the 
imaginary  unit  i  as  follows  : 

V—  a  =  V(—  l)a  =  V—  1  Va  =  ^Va. 

Whenever  a  pure  imaginary  number  occurs  in  any 
algebraic  work,  it  is  to  be  expressed  in  terms  of  the 
imaginary  unit  ^. 


.-A 

Karl  Friedrich  Gauss  (1777-1855)  is  called  by  general  agree- 
ment the  greatest  mathematician  of  modern  times.  In  1799  he 
published  a  proof  of  the  theorem  that  every  algebraic  equation  has 
a  root  of  the  form  a  +  bi.  He  introduced  the  symbol  /  to  denote 
V  —  1  and  was  the  originator  of  a  great  part  of  the  modern  theory 
of  numbers. 


QUADRATIC  EQUATIONS  331 

256.   Powers   of   /.     We  have  by  definition,  (V—  1)2 
=  —  1,  or  1*2  =  —  1 ;  therefore, 
i  =  V^T. 
z2  =  -  1. 

^3  =z  iH  =  (—  V)i  =  —  i. 
1^=  0*2)2=  (-1)2=  +1. 


iH  =  1  '  i  =  i. 


From  these  identities  it  is  inferred  that  any  even 
power  of  i  is  a  real  number,  namely,  +  1  or  —  1,  and  that 
any  odd  power  of  i  is  a  pure  imaginary  number,  namely, 
+  2  or  —  ^. 

ILLUSTRATIVE  EXAMPLES 


1.  Find  the  product  of  V—  4  and  V—  9. 
Solution.  \/31=V(£T)4=V^Vi  =  2i. 

v^  =  V(-  1)9  =  V-  1  V9  =  3  i. 

Hence,      v""^  V  -  9  =  (2 1)(3  i)  =  6  i^  =  6(-  1)  =  -  6. 

Note.  An  error  is  often  made  in  finding  the  product  of  two 
imaginary  numbers  by  an  incorrect  use  of  the  identity  y/a  Vb  =  y/ab. 
In  the  proof  of  this  identity  the  numbers  a  and  b  ,were  limited  to 
positive  numbers ;  the  expressions  V  —  a  and  V—  b  were  entirely 
meaningless.  This  error  is  avoided  by  observing  that  a  pure  imagi- 
nary number  is  to  be  expressed  in  the  form  ai,  where  a  is  a  real  number 
[§255]. 

2.  Simplify  — 

Solution.  _l__  =  l  =  i  =  _J_  =  _j. 

V3i      i     i^     -1 

3.  Simplify  -V^^  x  V^Tg  --  V^^5. 

Solution.     V32  x  V^3  ^  V^^  ^iy/2iVS 

iV5 


>/6     6  6 


332 


ELEMENTARY  ALGEBRA 


257.  Simplest  form  of  an  expression  containing  a  pure 
imaginary.  An  expression  which  contains  pure  imaginary- 
numbers  is  said  to  be  in  its  simplest  form  when  its  de- 
nominator is  a  rational  number  and  its  numerator  con- 
tains no  real  factor  under  a  radical  sign  which  can  be 
removed. 


EXERCISE  122 


Simplify 
1 


13. 


15. 


17. 


19. 


V- 

-4a2 

v^ 

-5 

V- 

-30 

i- 

«2 

"  3* 

V- 

-2-hV-3 

V-2 

V2 

4-V-3 

V2 


"J 


V-3 
4.    v^lV^^. 

6.     ^^, 
V3 

8.   v^V^^. 
1 


V-3a 

19 

V-30 

V-5 

14. 

v-|. 

16. 

u 

la 

V^a  +  V^ 

V-6 

9n 

V-2  +  3 

22.    t^. 


QUADRATIC  EQUATIONS  333 


1 
ill* 

23. 

24.    V-125. 

25.    V- 24x36x53. 

258.  Complex  numbers.  A  number  which  can  be  ex- 
pressed in  the  form  a  -h  5V—  1,  where  a  and  h  are  real 
numbers,  neither  one  of  which  is  zero,  is  called  a  complex 
number. 

Thus,  4+V—  1,  2— \/-5,  and  5  +  2V—  1  are  complex  numbers. 

259.  Conjugate  complex  numbers.  Two  complex  num- 
bers which  differ  in  the  sign  of  the  imaginary  unit  only 
are  called  conjugate  complex  numbers. 

Thus,  1  +  iy/2  and  1  -  iV2 ;  \/2  +  V^^a  and  V2  -  V^I^  ;  a  +  6i 
and  a  —  hi,  are  pairs  of  conjugate  complex  numbers. 

260.'  Sum  and  product  of  two  conjugate  complex  numbers. 

Let  a  +  hi  and  a  —  hi  be  any  two  complex  numbers ;  then : 

The  sum  =  (a  +  hi)  +  (a  —  hi)  =  2  a. 

The  product  =  (a  +  hi)  (a  -  i{)  =  a^  -  (6i)2 

=  a2  _  J2^-2  =  a2  _  52(_  1)=  a2  +  62. 

Hence  both  the  sum  and  the  product  of  two  conjugate 
complex  numbers  are  real  numbers. 

ILLUSTRATIVE   EXAMPLES 

1.    Find  the  product  of  V2  +  V^^  and  V3  +  V"^5". 
Solution.  V2  +  iV3 

VE  +  3i 

+  ^  VTO  +  i2Vl5 

V6  +  i(3+Vl0)-Vl5 
Therefore, 


334  ELEMENTARY  ALGEBRA 

Remark.  Observe  that  the  product  of  the  two  complex  num- 
bers is  a  complex  number. 

Q 

2 .  Rationalize  the  denominator  of  the  fraction = . 

V5  +  V-3 

Solution.  Since  the  product  of  two  conjugate  complex  numbers 
is  real,  we  multiply  the  numerator  and  denominator  of  the  given 
fraction  by  an  expression  conjugate  to  the  denominator,  and  have 

V5+V^     VS-V-a  5  +  3 

Note.  An  imaginary  number  is  sometimes  defined  as  "  any  even 
root  of  a  negative  number."  However,  although  V—  1,  for  exam- 
ple, is  evidently  not  a  real  number,  it  is,  nevertheless,  not  a  pure 
imaginary,  but  a  complex  number.      For,  it  may  be  verified  that 

(_L  +  A.,V    (±-±-i)\  (-^^±i)\  or  i-^-^iY 
VV2     V2   /      W2      \/2   /      \     V2     V2    /  V     V2      V2   / 

is  equal  to  —  1 ;   hence,  each  of  the  expressions  within  the  paren- 
theses is  a  fourth  root  of  —  1. 


EXERCISE  123 


Simplify 

16 

3. 
4. 
5. 
6. 
7. 
8. 

+  V-36. 
V3V-2. 

2. 

-  a(a  >  i 

r4)(V: 

-2  +  V- 

+  3V- 

• 

-3). 

1.    V- 

-Tkv 

(V-2-V- 
(l  +  V-5)2. 

9.  (2-hV-3)(2-V-3). 

10.  q  +  i^-^^i-iV^). 

11.  V— a-j-V— i. 

12.  (5  +  ^V3)-^(5-^V3). 


QUADRATIC  EQUATIONS  335 


2  2 

13.     = 14. 


-3  +  1  V-2h-V^ 

5 


15. 


V-lO-V-5 
261.  Quadratic  equations  with  complex  roots. 

ILLUSTRATIVE  EXAMPLE 

Solve  the  equation  2a;2  —  3a;4-5  =0. 
Solution  by  completing  the  square. 

2x2 -3a: +  5  =  0.  (1) 

Transforming  (1),              a:^  -  |  a:  +  |  =  0.  (2) 

Transposing,                               x^  —  ^  x  =  —  ^.  (3) 

Completing  the  square,  a:^  —  |-a:  +  ^  =  —  f  +  y^6=—  ^q-        (4) 

That  is,                                      (a:-|)2  =  _ll.  (5) 

Extracting  the  square  root,          ^c  —  ^  =± 4V—  31.  (6) 

Solving  (6),                                           ^=i±^El.  ^            (7) 
Solution  by  formulae. 


_  ^,  +  V'52  _  4  ac  -  6  -  V&2  _  4  ac 

xi  = ,  x^  = 

2a  2a 

Here  a  =  2,  6  =  -3,  c  =  5;  hence,  by  substitution  in  the  formulae 

we  find  the  roots  to  be 


-(-3)  +  V9-40  ^^^  -(-3)-V9340 
4  4 

which  when  simplified  are  equal,  respectively,  to 

3+^^=^  and  S^l^^Zl. 


EXERCISE  124 

Solve  the  following  equations,  which  have  either  pure 
imaginary  or  complex  roots : 

1.    2:24.1  =  0.  2.    ic2  +  4  =  0. 


336  ELEMENTARY  ALGEBRA 

3.  a;2+9  =  0.  4.  3^2^2  =  0. 

5.  2a;2  +  3  =  0.  6.  3a^  +  5  =  0. 

7.  5a:2-h2  =  0.  8.  15a;2  +  17  =  0. 

9.  a;2+2a;  +  2  =  0.  10.  2^2^43,.^  5  =  0. 

11.  2^2-62: +  11  =  0.  12.  a:2  +  5a;  +  7  =  0. 

13.  2a^-32;  +  2  =  0.  14.  32^2_3^+2  =  0. 

EXERCISE  125 

1.  A  man  bought  a  certain  number  of  oranges  at 
75  cts.  The  number  of  oranges  he  bought  was  three  times 
the  number  of  cents  he  paid  for  each  orange.  How  many 
oranges  did  he  buy  ? 

2.  The  side  of  one  square  is  three  times  the  side  of 
another  and  the  difference  of  their  areas  is  32.  What  is 
the  side  of  the  smaller  square  ? 

3.  If  the  edge  of  a  certain  cube  be  doubled,  the  area 
of  the  entire  surface  of  the  cube  will  be  increased  by 
72  sq.  in.     What  is  the  edge  of  the  cube  ? 

4.  A  dealer  sold  an  article  at  a  loss  of  f  6.25  and 
thereby  lost  as  many  per  cent  as  there  were  dollars  in  the 
cost.     What  was  the  cost  ? 

5.  Find  two  consecutive  integers  whose  product  is  56. 

6.  Find  two  consecutive  integers  whose  product  is 
462. 

7.  Find  a  number  whose  square  exceeds  100  times  the 
number  by  2684. 

8.  Two  odd  integers  differ  by  2  and  the  difference  of 
their  squares  is  56.     Find  the  integers. 

9.  The  quotient  of  two  numbers  is  2J  and  tfieir  prod- 
uct is  756.     Find  the  numbers. 


QUADRATIC  EQUATIONS  337 

10.  The  difference  of  the  squares  of  two  consecutive 
numbers  is  197.     Find  the  numbers. 

11.  The  sum  of  two  numbers  is  21,  and  their  product 
is  110.     What  are  the  numbers  ? 

12.  The  difference  of  two  numbers  is  4,  and  their  prod- 
uct is  45.     What  are  the  numbers  ? 

13.  The  difference  of  two  numbers  is  42,  and  their 
quotient  is  the  less  number.     What  are  the  numbers  ? 

14.  A  dealer  sold  an  article  for  $  39  and  thereby  gained 
as  many  per  cent  as  there  were  dollars  in  the  cost.  Find 
the  cost. 

15.  The  plate  of  a  looking-glass  is  18  in.  by  12  in. ;  it 
is  to  be  surrounded  by  a  plain  frame  of  uniform  width, 
whose  area  shall  be  equal  to  that  of  the  glass.  Required 
the  width  of  the  frame. 

16.  Find  three  consecutive  integers  the  sum  of  whose 
products  by  pairs  is  299. 

17.  If  a  body  be  thrown  vertically  upward  from  the 
ground  with  an  initial  velocity  of  32  ft.  per  second,  when 
will  it  be  at  a  height  of  7  ft.  ? 

Suggestion.  Use  the  formula  s  =  at  —  16  fi,  in  which  a  represents 
the  initial  velocity  and  s  the  height  at  the  end  of  t  seconds. 

18.  The  denominator  of  a  given  fraction  is  one  greater 
than  its  numerator ;  if  ^"^  be  added  to  the  fraction,  the 
sum  is  equal  to  the  reciprocal  of  the  given  fraction.  Find 
the  given  fraction. 

19.  The  difference  between  the  hypotenuse  and  base  of 
a  right-angled  triangle  is  6,  and  the  difference  between 
the  hypotenuse  and  altitude  is  3.     What  are  the  sides  ? 

20.  A  square  garden  is  surrounded  by  a  path.  The 
area  of  the  path  is  12,400  sq.  ft.  The  garden  is  290  ft. 
wider  than  the  path.     Find  the  area  of  the  garden. 


338  ELEMENTARY   ALGEBRA 

21.  A  field  containing  one  acre  is  in  the  form  of  a  rec- 
tangle I  as  wide  as  it  is  long.  The  field  is  enlarged  by 
adding  39,664  sq.  ft.  in  such  a  way  as  to  increase  length 
and  width  of  the  rectangle  an  equal  amount.  Find  the 
dimensions  of  the  enlarged  field. 

22.  From  the  point  of  intersection  of  two  straight 
roads  which  intersect  at  right  angles,  two  men,  A  and  B, 
set  out  simultaneously,  A  on  the  one  road  riding  at  the 
rate  of  12  mi.  per  hour,  and  B  on  the  other  walking  at  the 
rate  of  5  mi.  per  hour.  After  how  many  hours  will  they 
be  65  mi.  apart  ? 

23.  A  number  of  laborers  were  employed  to  do  a  piece 
of  work.  If  7  less  had  been  employed,  the  work  would 
have  taken  two  more  days.  If  28  men  had  been  em- 
ployed, the  work  would  have  been  done  in  20  days. 
How  many  laborers  were  employed  ? 

24.  A  gardener  planted  a  certain  number  of  trees  at 
equal  distances  apart,  and  in  the  form  of  a  square.  He 
found  on  finishing  the  planting  that  he  had  5  trees  to 
spare.  He  then  added  one  of  them  to  each  row  as  far  as 
they  would  go,  and  found  that  he  needed  10  trees  to  com- 
plete the  square.     How  many  trees  had  he  ? 

25.  Find  the  price  of  tea  per  pound  if  a  rise  of  10 
cents  in  the  price  per  pound  would  reduce  by  5  lb.  the 
quantity  obtainable  for  $15. 

26.  What  is  the  price  of  eggs  per  dozen,  if  a  fall  of 
2  cents  in  the  price  would  increase  by  one  the  number 
of  dozen  obtainable  for  i6.84  ? 

27.  Two  trains  travel  without  stopping  between  two 
stations  m  miles  apart.  One  train  goes  a  miles  an  hour 
faster  than  the  other  and  takes  h  hours  less  time  for  the 


QUADRATIC  EQUATIONS  339 

journey.     Find   the  speed  of   each  train.     What   is   the 
speed  of  each  train  if  m  =  40,  a  =  10,  and  h  =  ^? 

28.  A  man  bought  a  certain  number  of  cows  for  flSOO. 
He  sold  5  less  than  the  whole  number  of  cows  for  1 20  a 
head  more  than  they  cost  him  and  made  $100  by  the 
transaction.     How  many  cows  did  he  buy  ? 

29.  A  merchant  sold  7  doz.  fresh  eggs  and  12  doz. 
storage  eggs  for  $5.81,  and  found  that  he  had  sold  1  doz. 
more  fresh  eggs  for  $2.10  than  he  had  of  storage  eggs 
for  $1.40.     Required  the  price  of  each  kind  per  dozen. 

30.  A  merchant  draws  a  certain  quantity  of  vinegar 
from  a  full  cask  containing  63  gallons.  Having  filled  up 
the  cask  with  water,  he  draws  the  same  quantity  as 
before.  He  then  finds  that  the  cask  contains  ^g  the 
original  quantity  of  vinegar.  How  many  gallons  did 
he  draw  each  time  ? 

31.  A  sum  of  $30,000  is  subject  to  an  inheritance  tax 
of  a  certain  per  cent,  then  to  a  percentage  for  fees  at  a 
rate  one  half  per  cent  greater  than  that  of  the  inheritance 
tax.  When  the  tax  and  fees  are  deducted  there  remains 
$  27,504.     What  are  the  two  rates  ? 

262.  Utility  of  the  extension  of  the  meaning  of  the  word 
number.  The  extension  of  the  meaning  of  the  word  num- 
ber so  as  to  include  under  the  term  such  expressions  as 
V—  2  and  2  +  V—  3  renders  possible  a  greater  generality 
in  the  statement  of  algebraic  principles  and  results.  As 
an  illustration  of  this,  the  statement  that  except  when 
l^  —  4:ac  is  negative,  the  quadratic  equation  ax^  -\-bx-\-  c=0 
admits  of  solution  [section  248,  note],  is  replaced  by  the 
general  statement,  everi/  quadratic  equation  has  two  roots. 


340  ELEMENTARY  ALGEBRA 

263.  Nature  of  the  roots  of  ax^  +  bx+c  =  0.  In  this 
equation  the  a,  6,  and  c  represent  rational  numbers.  The 
roots  of  ax^  -\-bx-\-  c=0  have  been  found  to  be, 


a?,  = -^—- and  x^  = 

An  examination  of  these  formulae  leads  to  the  follow- 
ing important  principles : 

I.   When  6^  _  4  ac  is  positive  and  not  equal  to  zero. 

In  this  case, 

1.  The  roots  are  real  and  unequal. 

Thus,  in  the  equation  Oa:^  —  18a;  +  7  =  0  the  expression  6^  —  4  ac 
=  72,  and  the  roots,  1  ±  J"^)  are  real  and  unequal. 

2.  If  52  —  4  ac  is  a  perfect  square,  the  roots  are  rational ; 
and,  conversely,  the  roots  are  rational  only  when  5^  —  4  ac 
is  a  perfect  square. 

Thus,  in  the  equation  2ar2  +  5a;  —  3  =  0,  the  expression  b^  —  iac 
=  49,  and  the  roots,  —  3  and  -^ ,  are  rational. 

3.  If  b^  —  4:ac  is  not  a  perfect  square,  the  roots  are 
conjugate  quadratic  surd  expressions. 

Thus,  in  the  equation  9  x^  —  18  a;  +  7  =  0,  the  expression 
b^  —  iac  =  72,  and  the  roots  1  +  J  V2  and  1  —  J  V2  are  conjugate 
quadratic  surd  expressions. 

4.  If  c  is  0,  one  root  is  zero,  and  the  second  root  is 
rational. 

Thus,  in  the  equation  a:*  —  3  a:  =  0,  the  roots  are  zero  and  3. 

Note.  If  a  and  c  have  opposite  signs,  6*  —  4  ac  is  necessarily  posi- 
tive and  the  roots  are  always  real. 

II.   When  J^ - 4 acis  equal  to  zero. 
In  this  case, 


QUADRATIC  EQUATIONS  341 

1.  The  roots  are  rational  and  equal. 

Thus,  in  the  equation  9x^  —  Qx-^l  =  0,  the  expression  b^  ~  i^ac 
=  0,  and  the  roots  are  ^  and  J. 

2.  The  polynomial  aa^  +  5a;  +  ^  is  a  perfect  square. 
Thus,  9a;2-6a:  +  l  =  0  may  be  written  (3  ar  -  1)2  =  0. 

3.  When  c  is  zero,  then  b  is  also  zero,  and  both  roots 
are  zero. 

Thus,  if  62  _  4  ac  =  0,  and  c  =  0,  then  b^  =  0,  or  5  =  0 ;  and  the 
roots  of  the  equation  are  0  and  0. 

III.   When  6^  —  4  ac  is  negative. 
In  this  case, 

1.  The  roots  are  not  real. 

Thus,  in  the  equation  3a;2  —  2ar  +  4:  =  0,  the  expression  b^  —  iac 

=  ^  44  and  the  roots  are  ~ —  and     ~     ~ — ,  which  are  not 

3  3         ' 

real. 

2.  When  b  is  zero,  the  roots  are  pure  imaginary  numbers. 

Thus,  in  the  equation  a:2  +  4  =  0,  the  expression  b^  —  iac  =  —  i 
and  b  is  zero.  The  roots  are  2  i  and  —  2  e,  which  are  pure  imaginary 
numbers. 

3.  When  b  is  not  zero,  the  roots  are  conjugate  complex 
numbers. 

Thus,  in  the  equation  3a;2  —  2a;  +  4  =  0,  which  is  the  equation 
given  in  1,  the  roots  are  seen  to  be  conjugate  complex  numbers. 

Note.  The  statement  that  when  b^  —  4:ac  =  0  the  polynomial 
ax^  -\-  bx  -\-  c  is  a.  perfect  square,  may  be  proved  as  follows : 

Given  ^2  -  4  ac  =  0. 

Transposing,  62  —  4  qc. 

Therefore,  b  =±2y/aVc. 

Substituting,  ax^  -{- bx  +  c  =  ax^  ±  2VaVcx  +  c 

=  (ar Va  ±  Vc)2. 


a42  ELEMENTARY  ALGEBRA 

EXERCISE  126 

Calculate  for  each  of  the  following  equations  the  value 
of  h^  —  4  ac  and  determine  from  principles  of  section  263 
the  nature  of  the  roots.  The  principles  should  not  be 
memorized,  but  the  reason  for  each  statement  made  should 
be  clearly  understood. 

1.    a:2H-3a:-4  =  0. 

3.    5a:2  =  0. 

5.    3a;2  +  2a;  +  5  =  0. 

7.    ic2-a;-l  =  0. 

9.    9a:2-12a:  +  4  =  0. 
11.    |a;2-f^  +  ^Y^  =  0. 
13.    2a;2_3^_20  =  0. 
15.    a^2_^2«rr  +  a2+52=o. 

264.   Factors  of  ax^+hx-\-  c.     By  definition,  a  root  of 

an  equation  is  a  number  which,  when  substituted  for  the 
unknown,  reduces  the  equation  to  an  identity  ;  that  is, 
satisfies  the  equation. 

Let  Xj  represent  a  root  of  the  equation, 

ax^ -\- hx -\-  c  =  0.  (1) 
Substituting  x-^  for  x  in  (1), 

ax^  +  6^1  +  c  =  0.  (2) 

Solving  (2)  for  c,                              c  =  -  ax^^  -  hx^  (3) 
.Substituting  the  value  of  c  in  (1), 

ax^  +  hx  ■\-  c  =  ax^  +  bx  -  ax^  -  bx^  (4) 

=  a(x^-x,^)+b(x-x^  (5) 

=  (x  -  xi)  (ax  +  axi  +  b)  (6) 

From  the  foregoing  it  may  be  inferred  that : 
if  x^  is  a  root  of  the  equation  a^  +  5a;  -|-  c  =  0,  the  polyno- 
mial aoi^  4-  ia;  +  c  is  exactly  divisible  by  x—  x^ 


2. 

2a:2_3_0. 

4. 

a;2  +  1  =  0.  ' 

6. 

3a^-2rr-2  =  0. 

8. 

a^-{-x-{-l  =  0. 

10. 

fa^-|a;  +  |  =  0. 

12. 

4.9a:2-7.35a;-22.05  =  0. 

14. 

169a;2+442^4.289  =  0. 

16. 

a:2_2aa:  +  a2  +  52=0. 

QUADRATIC  EQUATIONS  34$ 

A  second  root  of  ax^  +  62:  +  c  =  0  is  obtained  by  equat- 
ing the  second  factor  of  ax^  -\-hx-^  c^  namely,  ax  +  ax^  +  6, 
to  0  and  solving  for  x  [see  section  117]. 

Thus,                                  ax  +  a^i  +  6  =  0.  (1) 

Solving  (1)  for  x,      ,  x  —  —x^ •  (2) 

Representing  the  root  —  x^  —  by  x,^  we  have, 

Transposing,  x^-\-  x^  = (^) 

Identity  (4)  is  in  agreement  with  (I)  of  section  249. 
The  expression  ax  +  ax^  +  h  may  be  written  a[x-\-x^-\--\ 

or  substituting  —  x^  for  a;^  +  - ,  we  have  a{x  —  x^.  There- 
fore, the  polynomial  a'39-  -\- hx -\-  c  is  identically  equal  to 
a{x  —  Xy){x  —  x^  where  x^  and  x^  are  two  roots  of  the 
quadratic  ax?'  +  6a;  +  c  =  0. 

265.  Number  of  roots  of  a  quadratic.  Let  x^  be  a  root 
of  the  quadratic  equation 

ax^  -\- hx -\-  c  —  ^  \ 

that  is,  of  the  equation 

a{x  -  x^{x  -  x^=^, 

in  which  x-^  and  x^  are  two  roots  of  the  quadratic.  Ob- 
serve that  the  existence  of  at  least  two  roots  was  shown 
in  section  248. 

Substituting  x^  for  x  in  the  given  equation,  a(x^  —  x^^ 
(a;g  —  x^)  =  0.  From  this  identity  it  is  evident,  since  a  is 
not  zero,  that  either  x^  —  x^  =  0  or  x^  —  x^^O;  that  is,  x^ 
is  equal  to  either  x^  or  x^;  and,  therefore,  that: 

Every  quadratic  equation  has  two  and  only  two  roots. 


344  ELEMENTARY  ALGEBRA 

266.  Quadratic  with  given  roots.  The  quadratic  equa- 
tion whose  roots  are  x^  and  x^  is  a(^x  —  x^^(^x  —  x^)  =  0  [see 
section  264].  In  this  equation  a  may  have  any  constant 
value. 

Let  the  two  equations  ax^  +  bx  +  c  =  0  and  mx^  +  nx  +  p  =  0  have 
the  same  roots  x^  and  x^    We  may  write  [see  section  249]  : 

b] 


^1  +  ^2  =  - 
a 


and 


•^1  ~f"  "^2 

m 
m 


From  these  identities,  —=-  =  —.     Therefore  : 
m     n     p 

Two  quadratic  equations  with  the  same  roots  have  their  cor- 
responding coefficients  proportional ;  and  conversely,  if  two 
quadratic  equations  have  their  corresponding  coefficients  pro- 
portional^ they  have  the  same  roots. 

ILLUSTRATIVE  EXAMPLE 

P^ind  the  simplest  form  of  a  quadratic  equation  whose 
roots  are  —  |  and  |. 

Solution.     The  required  equation  is  a{x  +  J)(a:  —  ^)=  0. 

That  is,  a(3^)(5^)  =  0. 

Letting  a  =  15,  (3  a:  +  2)(5 x  -  3)  =  0. 

Expanding,  15  a;^  +  a;  -  6  =  0. 

EXERCISE  127 

1.  By  inspection,  arrange  the  following  equations  in 
groups  so  that  those  in  any  group  shall  have  the  same  roots : 

5  2:2  _  15  a;  ^.  10  =  0.  10  a^  -  15  a:  -  10  =  0. 

a:a~§£-l  =  0.  m^-Smx-{-2m=0, 

2 


QUADRATIC  EQUATIONS 
Find  the  quadratic  whose  roots  are : 


345 


2.    —  5  and  6.  3.    |  and  f 

5.    —  t  and  X. 


4.    -f  and  -f 


e.    1±;^  and  3Ll^ 


7.    — r—  and  — -— ■ 


3_    l  +  2V-2^^^1-2V-2_ 


„     2  +  3i       .  2-3i 

267.  Graph   of   a  polynomial   of    the    second    degree. 

Graphs  of  linear  functions  of  one  variable  and  of  linear 
equations  of  two  unknowns  were  treated  in  Chapter  IX, 
We  shall  now  consider  the  graphs  of  certain  quadratic 
functions. 

268.  Graph  of  x^  —  2  x.  Representing  the  polynomial 
x^  —  2xhj  1/  ;  then, 

i/^x^-^2x.  (1) 

We  construct  a  table  of  the  corresponding  values  of  x  and  y  by 
arbitrarily  assigning  values  to  x  and  calculating  the  corresponding 
values  of  y  from  equation  (1).     The  table  follows : 


X 

-4 

-3 

-2 

-1 

0 

1 

2 

3 

4 

5 

6 

X2 

16 

9 

4 

1 

0 

1 

4 

9 

16 

25 

36 

-2x 

8 

6 

4 

2 
3 

0 

-2 

-4 

-6 

-8 

-10 

-12 

y 

24 

16 

8 

0 

-1 

0 

3 

8 

16 

24 

Plotting  the  points  (x,  y)  as  given  in  the  table  and  drawing  a 
smooth  curve  through  these  points,  we  have  the  required  graph  of 
the  polynomial  3^-  2XfM  indicated  in  the  figure,  page  346. 


346 


ELEMENTARY  ALGEBRA 


From  the  graph  of  the  poly- 
nomial x^  -2x;  that  is,  from 
the  graph  of  the  equation 
y  =  z^  —  2x,  the  roots  of  the 
quadratic  equation  a:^  -  2  a:  =  0 
may  be  found  by  inspection. 
Evidently,  those  values  of  x 
which  make  y  equal  to  zero 
are  the  roots  of  the  equation 
x^  —  2x  =  0,  for  they  satisfy  the 
equation.  The  points  on  the 
graph  for  which  y  =  0  are  those 
common  to  the  graph  and  the 
a:-axis ;  for  the  ordinate  y  oi  a, 
point  is  zero  only  when  the 
point  is  on  the  x-axis.  The  roots 
of  the  equation  a:^  —  2  a:  =  0  are, 
therefore,  the  a:(abscissas)  of 
the  points  in  which  the  graph 
of  the  polynomial  x^  —  2x  in- 
tersects the  X-axis.  These 
points  of  intersection  are  x  =  0 
and   X  =  2j  and    the    roots   of 


the  equation  ar^  -  2a;  =  0  are  0  and  2. 


269.   Graphof  x2-A:-f  1. 

Let  y  =  x^  —X  -h  1. 

A  table  of  corresponding  values  of  x  and  y  is  as  follows : 


X 

-4 

-3 

-2 

-1 

0 

\ 

1 

2 

3 

4 

6 

^ 

16 

9 

4 

1 

0 

\ 

1 

4 

9 

16 

25 
-4 

-a;-f-l 

6 

4 

8 

2 

1 

\ 

0 

-1 

-2 

-3 

V 

21 

13 

7 

3 

1 

\ 

1 

3 

7 

13 

21 

Plotting  the  points  (a:,  y)  as  given  in  the  table  and  drawing  a 
smooth  curve  through  these  points,  we  have  the  required  graph  of 
the  polynomial  a;^  —  a;  -h  1,  as  in  the  figure,  page  347. 


QUADRATIC  EQUATIONS 


347 


We  observe  that  the  graph  of  the  equation  y  =  x^  —  x  +  1  does 
not  intersect  the  axis  of  x.  This  indicates  that  the  quadratic 
equation  a;^  —  x  +  1  =  0  has  no  real  root.     The  roots  of  this  equation 


are 


1±  V- 


that  is,  conjugate  complex  numbers.     In  general, 


when  the  graph  of  a  polynomial  ax^  +  bx  -^  c  does  not  intersect 


m 


the  X-axis,  the  roots  of   the   equation  ax^  +  bx  -\-  c  =  0  are   either 
pure  imaginary  numbers  or  conjugate  complex  numbers. 


270.    Graph  of  a  system  of  two  simultaneous  equations. 
Given  the  system  of  two  simultaneous  equations, 


(1) 
(2) 


In  section  192  we  learned  that  the  graph  of  the  linear  equation 
y  =  2x  is  a  straight  line.  It  is,  therefore,  necessary  to  find  the 
codrdinates  of  two  of  its  point*  only  in  order  to  plot  the  straight 


348 


ELEMENTARY  ALGEBRA 


line.  We  observe  from  equation  (2)  that  the  points  (0,  0)  and 
(3,  6)  may  be  taken.  The  corresponding  values  of  x  and  y  in 
equation  (1);  that  is,  of  x  and  a:^,  are  given  in  the  preceding  table. 

The  graphs  of  equations  (1)  and 
(2)  are  seen  in  the  accompanying 
figure. 

The  real  solutions  of  equations 
(1)  and  (2)  are  the  coordinates 
of  the  points  of  intersection  of 
their  graphs ;  for  the  coordinates 
of  these  points  satisfy  both  equa- 
tions, and  the  coordinates  of  any 
point  not  on  both  graphs  do 
not  satisfy  the  equations.  The 
straight  line  y  =  2  a:  intersects 
the  graph  of  y  =  a;^  in  the  points 
(0,  0)  and  (2,  4).  The  two  solu- 
tions of  the  simultaneous  equa- 
tions y  =  x^  and  y  =  '2x  are, 
therefore,  (0,  0)  and  (2,  4>. 

When  the  graphs  of  two  equar 
tions  do  not  intersect,  their  solu- 
tions involve  pure  imaginary  num- 
bers or  complex  numbers. 

The  graphs  of  y  —  x"^  and 
y  ■=2x  also  show  by  their  intersections  the  roots  of  the  equation 
a:2  -  2  a;  =  0. 

Thus,  the  ordinate  of  any  point  on  the  straight  line  whose- 
equation  is  y  =  2  a:  is  equal  to  twice  its  abscissa,  and  the  ordinate 
of  any  point  on  the  graph  of  y  —  x"^  is  equal  to  the  square  of  its 
abscissa.  At  a  common  point  the  ordinate  is  equal  to  twice 
its  abscissa  and  also  to  the  square  of  its  abscissa.  The  abscissa  of 
a  common  point,  therefore,  satisfies  the  equation 

a;2  =  2  a:,  or  a:2  -  2  a;  =  0. 

In  like  manner  the  real  roots  of  aa;*  4-  &a:  +  c  =  0  are  the  abscissas 
of  the  points  common  to  the  graphs  of  the  equations 


L_                            Y 

, 

L                      7 

f            I 

4               t   j- 

4              ^  ' 

1-            ^  7- 

'  t 

aJ 

X         t' 

4          H 

4-       l|- 

t     t 

X  JL 

)^2L 

-^t 

x-j               7"o                       X 

t 

7 

Y'^ 

y  =  Qx^  and  y  =  -  &x  -  c. 


QUADRATIC  EQUATIONS  349 

EXEBCISE  128 

1.  Construct  the  graph  of  the  equation  y  =  qc^  —  1. 
From  the  resulting  graph  determine  the  roots  of  the  equa- 
tion a;2  —  1  =  0. 

2.  Construct  with  respect  to  the  same  axes  of  reference 
the  graphs  of  y  =  2x^—1  and  y  =  —  ^x  +  l.  Estimate 
from  the  figure  the  values  of  x  and  y  which  satisfy  both 
equations.  Also  obtain  approximately  the  roots  of 
2x^-\-Sx-2  =  0. 

3.  Construct  with  respect  to  the  same  axes  of  reference 
the  graph  of  y -\-2x^ -Sx -9  =  0  and  y-\-x~S  =  0.  Es- 
timate from  the  figure  the  values  of  x  and  y  which  satisfy 
both  equations.  Also  obtain  approximately  the  roots  of 
x^-2x-^  =  0. 

4.  Construct  the  graph  of  x^—  2.  Estimate  from  the 
figure,  correct  to  one  decimal  place,  the  value  of  V2. 

5.  Plot  2y—Sx  =  6  and  2x'\-l=4:y —  4:y\  using  the 
same  axes,  and  estimate  from  the  graphs  the  solutions  of 
the  equations. 

6.  Graph  ^  =  1 +  32^. 

7.  Construct  with  respect  to  the  same  axes  of  reference 
the  graphs  oi  y  =  2x'^  -{-1  and  y  =  x^  -\-S  x  —  1.  Estimate 
from  the  figure  the  values  of  x  and  y  which  satisfy 
both  equations.  Also  obtain  from  the  figure  the  roots  of 
2^-Bx-{-2  =  0. 

8.  Construct  with  respect  to  the  same  axes  of  reference 
the  graphs  of  x^-\-y-5  =  0  and  y^-\-  Sy  —  2x=0.  Es- 
timate from  the  figure  the  real  values  of  x  and  y  which 
satisfy  both  equations. 


CHAPTER  XIII 
SYSTEMS   OF   QUADRATIC  EQUATIONS 

271.  Systems  of  two  equations  in  two  unknowns.     The 

elimination  of  one  of  the  unknowns  from  two  equations  of 
the  second  degree  in  two  unknowns  does  not,  in  general, 
lead  to  a  quadratic  equation  in  one  unknown.  Certain 
special  systems  of  two  equations,  however,  neither  one  of 
which  is  of  higher  degree  than  the  second,  are  of  frequent 
occurrence  and  lead  to  quadratic  equations  in  one  un- 
known.    Such  systems  may  be  solved  by  the  methods  of 

preceding  chapters. 
• 

272.  A  quadratic  and  a  linear  equation.  A  system  con- 
sisting of  a  quadratic  and  a  linear  equation  may  always 
be  solved  by  substitution.  The  method  is  indicated  in 
the  illustrative  examples  which  follow. 

ILLUSTRATIVE  EXAMPLES 
,.2 


Solve  the  system    \     „      .  ^  „ 


a) 

(2) 

Solution.     Solving  (2)  f  or  y,                 y  =  -Sx  +  13.  (8) 
Substituting  in  (1)  the  value  of  y  from  (3), 

a;a  +  2a:(-3a:+13)  =  33.  (4) 

Simplifying  (4) ,                  5  a;^  -  26  a: + 33  =  0.  (6) 

Factoring  (5),               (x  -  3)(5  a;  - 11)  =  0.  (6) 

From  (6),                                                    ar  =  3.  (7) 

Also  from  (6),                                          ^  =  ^'  (S) 

Substituting  3  for  a:  in  (3),                     y  =  4,  (O) 
One  solution  of  (1)  and  (2)  is,  therefore,  (3,  4) 

Substituting  ^  for  a:  in  (3),                   y  =  ^.  (10) 

The  second  solution  of  (1)  and  (2)  is,  therefore,  (-y-,  ■^). 

S50 


SYSTEMS  OF  QUADRATIC  EQUATIONS  351 

Check.     Substituting  3  and  4  for  x  and  y,  respectively,  in  (1)  . 

9  4-  24  =  33,  (11) 

which  is  an  identity. 

Substituting  JJ-  and  -^  respectively,  in  (1) 

,.,.         .,      .  W  +  W  =  ^^'  (12) 

which  IS  an  identity. 

The  values  of  y  were  obtained  from  (3),  which  is  another  form  of 

(2);  it  is,  therefore,  unnecessary  to  substitute  in  (2), 

2.    Solve  the  system 

2x^-Sxi/-\-  f-Bx  +  l  ^  -4:=0,  (1) 

4a:+3^  +  l=0.  (2) 

Solution.     Solving  (2)  for  y,  y  =  -  ijLtl        (3) 

Substituting  in  (1)  the  value  of  y  from  (3), 
2x«+3.(i^l)  +  (i^)'-5.-7(l^)-4  =  0.  (4) 

Simplifying  (4),                               5  a;^  -  8  a:  -  4  =  0.  (5) 

Factoring  (5),                               (x  -  2) (5  a:  +  2)  =  0.  (6) 

From  (6),                                                            '     a:  =  2.  (7) 

Also  from  (6),                                                             ^  =  -  f •  (8) 

Substituting  2  for  a:  in  (3),                                   2^  =  -  3.  (9) 
One  solution  of  (1)  and  (2)  is,  therefore,  (2,  —  3). 

Substituting  -  ^  ior  x  in  (3),                              !/ =  i'  (10) 
The  solutions  of  (1)  and  (2)  are,  therefore,  (2,  -  3)  and  (-  |,  •^), 

which  solutions  should  be  verified  by  substituting  in  (1)  the  values 

of  X  and  y  as  found. 

EXERCISE    129 

Solve  the  following  systems,  and  check  each  solution: 

*la;  =  l.  '    [^  —  x=:  0. 


a; +  2  3/ =  17.  '    \x-^  =  ^. 

\xy  =  2, 


3ir2_^2aj  +  y-ll  =  0, 
2x-y-\-l  =  0,  ^ 


352 


ELEMENTARY  ALGEBRA 


j6a;2^  +  4=0, 
''    I5a:-5^=21. 

-I 


6  a;y  +  4  =  0,  ^     f  a;^  _  ^  =  ^2  _|.  ^.^ 

(^  +  l)-(2;-l)  =  0. 
1 


10. 


11. 


=  8, 
xy 

1  +  2^17. 
a;     y 


12. 


13 


a;2_43/2  =  4, 
2a:  +  3«/  +  4  =  0. 

(2;-l)2  +  (y_3)2  =  25, 


fa;2_^2^3^^2«/-10  =  0, 


14. 


15. 


16. 


rr2  _  2  a;?^  +  «/2  -f  2  a;  -  2  ^  =  0, 
a;+^-2  =  0. 

2rr2-3a:^+4?/2_2a;  +  3^-6  =  0, 
5a:  +  4«/— 1  =  0. 

3a;2+2a;-3^~2  =  0, 
3a;  +  ^-4  =  0. 


273.  Two  quadratic  equations,  one  of  which  is  homogene- 
ous. When  one  of  the  given  quadratic  equations  is  homo- 
geneous, the  given  system  can  be  replaced  by  two  systems 
each  of  which  is  of  the  type  considered  in  section  272. 
The  method  of  solution  is  as  follows  : 


ILLUSTRATIVE  EXAMPLE 


x^-hx  +  Qy-lS  = 


2x^-xy-15y^  =  0,  (1) 

0.  (2) 


SYSTEMS  OF  QUADRATIC  EQUATIONS  353 

Expressing  the  homogeneous  polynomial  2  x^  —  xy  —  15  y^  a,s  the 
product  of  two  linear  factors,  equation  (1)  may  be  replaced  by 

(x-3y)(2x  +  5y)=0.  (3) 

Since  any  solution  common  to  equations  (2)  and  (3)  must  satisfy 
either  x  —  3y  =  0,  or2a;  +  5?/  =  0,  we  may  consider  first  those  solu- 
tions of  the  given  systems  which  satisfy 

x  -  3  y  =  0,  (4) 

and  a:2  +  a;  +  6  3/  -  18  =  0.  (2) 

Afterwards  those  that  satisfy  2a;+5y  =  0,  (5) 

and  a:2  +  x  +  6  y  -  18  =  0.  (2) 

Substituting  in  (2)  the  value  of  x  from  (4)  and  simplifying, 

y^  +  y-2  =  0.  (6) 

Solving  (6),  y  =  1;  also,  y  =  —  2. 

From  (4),  when  3/  =  1,  a;  =  3  ;  when  y  =  —  2,  a:  =  —  6. 
Therefore,  the  solutions  of  (4)  and  (2)  are  (3,  1)  and  (-6,  —  2). 
Substituting  in  (2)  the  value  of  y  from  (5)  and  simplifying, 

5x2 -7a; -90  =  0.  (7) 

Solving  (7),  x  =  5 ;  also  a:  =  —  ^-. 

From  (5),  when  a;  =  5,  ?/  =  —  2 ;  when  x  =  —  ^,  y  =  |-|.. 

Therefore,    the    solutions    of    (5)    and    (2)    are    (5,    —  2),  and 

(-¥.|f)- 

Therefore,  the  solutions  of  the  given  system  are  (3,  1),  (—6,  —2), 
(5,  -  2),  and  (-1/-,  |f). 

EXERCISE    130 

Solve  the  following  systems  : 

a;2_y2^0,  (X^-4:  2/^  =  0, 

3a:2  +  7a;?/  =  0,  (2x^  +  i/^+4:i/-2S  =  0, 

x^z-hlx  +  l y  +  49  =  0.  *•    13  0^2 +  80:3/ -3  3/2  =  0. 
a;2  4-  3  rry  -  5  ?/2  =  0,  f  2  2^2  +  7  a;^/  -  3  «/2  =  23, 


[2/ 


^-\-4:X2/-14:X-24:  =  0.  [d  X^  -  12  Xt/ -{- 4:  t/^  =  0, 


354  ELEMENTARY  ALGEBRA 

274.  Two  equations  without  terms  of  the  first  degree. 

A  system  of  two  simultaneous  quadratic  equations,  neither 
one  of  which  contains  terms  of  the  first  degree,  and  in 
which  the  constant  terms  do  not  reduce  to  zero,  may  be 
solved  as  in  the  following  : 

ILLUSTRATIVE  EXAMPLE 

Solve  the  system    (^  ^  "^  ^  ^^  +  ^' 7  ^'  ^^^ 

•^  1  3  a^  +  7  a:?/  +  2  3/2  =  2.  (2) 

Multiplying  both  members  of  (1)  by  2  so  that  the  constant  term 

in  the  resulting  equation  shall  be  the  same  as  that  in  (2), 

ix^  +  4:  xy  +  2  y^  =z  2.  (3) 

From  (2)  and  (3)  by  subtraction,     x^  —  S  xy  =  0.  (4) 

The  given  system  can  now  be  replaced  by  the  equivalent  system, 

x(x-Sy)  =  0,  (5) 

2x^+2xy  +  y^  =  l.  (1) 

The    system    (5)    and    (1)    can    be    solved    by  the  method  of 

section  273. 

Solving,  we  obtain  the  following  solutions : 

(0,1),(0,  -l),(f,^),(-f. -J). 

Remark.  If  the  constant  term  in  one  of  the  given  equations  is 
not  a  multiple  of  the  other,  the  equations  may  be  multipUed  by  such 
numbers  as  will  make  the  constant  terms  equal. 


U2- 


7. 


EXERCISE  131 

Solve  the  following  systems  : 

+  42^^/  +  3^2=2,       2  1^2-32:^  +  2^2^3^ 

4a:y+3^/2=12.         *  U24.  2  2:y  -  3  ^^2^  3, 

1^:2 -h  3  1/2  =  21,  g  [2a:2_3a:^/  +  ^/2  =  3, 

1 2:2  + 2  a:?/- 3^/2=  15.       *  I  a^  +  a:!/ +  i/2  =  7. 

-3^:2/4-3^/2  =  12,  f  8x2  + 11  a:i/+ 8^2  =  12, 


fa:2-3 
12  2:2- 


2a;a-a:y +  4^2^  16,  Il5a:2+18a;2/  +  12y2=20 


SYSTEMS  OF  QUADRATIC  EQUATIONS  355 

275.   Particular  systems  of  equations. 

I.  When  the  sum  and  product  of  two  unknowns  are  given, 
either  of  two  special  methods  of  solution  may  be  em- 
ployed ;   thus  : 

Solve  the  system    \    ^^       '  )^ 

•^  \x7/=n.  (2) 

Solution  1.  From  section  249  the  roots  of  the  quadratic 
x^  —  mx  -{-  n  =  0  are  two  numbers  whose  sum  and  product  are  equal, 
respectively,  to  m  and  n.  The  roots  of  the  equation,  therefore,  satisfy 
the  given  system. 

Solving  the  quadratic  x^  —  mx  +  n  =  0,  we  have, 


m  ±  y/m^  —  4  n 


Therefore  the  two  solutions  of  (1)  and  (2)  are 

(m  +  y/m^  —  4  w    m  —  Vm^  —  4  n\       ,  fm  —  Vm^  —  4  n   m  +  Vm^  —  4  n  \ 

2             '             2            )  ^"^    \           2            '             2  / 

Solution  2.        Squaring  (1)  x^  +  2  xy  +  y^  =  m\  (3) 

Multiplying  (2)  by  4,                                  4  xy  =  4  n.  (4) 

Subtracting  (4)  from  (3),         x"^ -2  xy  ^  y^  =  ni^  -  ^  n.  (5) 

That  is,                                                 ix  -yy  =m^-4:  n.  (6) 

Therefore,                          x-y  =±^1x1^  -  4  n.  (7) 

Adding  (1)  and  (7),            2x  =  m±  Vm^  -  4  n.  '     (8) 

Subtracting  (7)  from  (1),  2y  =  m^  \/m^  -  4  n.  (9) 

mi       i           r           .    m  +  Vm^  —  4  n        .    m  —  Vm*  —  4  n 
Therefore,  when  ar  is  — ■ ,  y  is , 


J     ,           .    m  —  Vm^  —  4  n       •    w  +  V/n^  +  4  n 
and  when  x  is ,  y  is . 


EXERCISE   132 

Solve  the  following  systems : 
*     U?/  =  1.  '    U^  =  12. 


356 

ELEMENTARY  ALGEBRA 

3.     i 

la:^  =  -28. 

4. 

^x+y  =  i, 

i^^  =  A- 

5. 

6. 

a;  H-  y  =  3  a, 

a^y  =  2  a2. 

7. 

D  a 
,  ly  =  1. 

8. 

^xy  -\-  a^  =  0, 

9. 

^^     3-2a' 
l-5a 

10. 

'      ,           2a+  36 
^  +  ^  =  2J-3a' 

26- 16a 
"^=3a-2J- 

II.     TFAeTi  the  difference  and  product  of  two  numbers  are 
given,  the  following  will  illustrate  a  method  of  solution: 


Solve  the  system    |       _  i  o   ' 


(1) 

(2) 


Solution.      Introducing  an  auxiliary  number  2,  defined  by  the 
identity  y  =  —  2,  the  given  equations  may  be  replaced  by 


a:  +  z  =  1, 
0:2;  = -12. 


(3) 
(4) 


The  system  (3)  and  (4)  now  belongs  to  class  I  and  may  be  solved 
by  the  methods  employed  therein,  the  solutions  being 


ar  =  4,  2 


3,  and  a;  =  —  3  and  2  =  4. 


Substituting  —  y  for  2,  the  solutions  of  the  given  system  are  (4,  3), 
(-3,-4). 

EXERCISE   133 

Solve  the  following  systems  : 


\xy  +  Q  =  (i, 
1 8  a;t^  +  1  =  0. 


2.    1^-^^ 
Uy=3. 


SYSTEMS  OF  QUADRATIC  EQUATIONS         357 


I  xy  =  3. 


a6  ~      I  xy 


xy-\-l  =  0. 

3a-  26 

X-  y  = =-, 

a  —  0 

35 


10. 


xy  = 


a  —  h 


a+  2 

^2g-  2a2  +  2a8 
"^^  2  a  -  3 


Solve  the  system        { 


III.      When  the  mm  of  the  squares  and  the  product^  mm, 
or  difference  of  two  numbers  are  given.     Typical  systems  are : 

ic2+3/2=5(l)  ^^^2  =  5(1)  2^^^2=5(1) 

xy  =  2  (2)  2:+«/=3     (3)  a:  -  y  =  1     (4) 

^2  +  2/^  =  5,  (1) 

^y=2.  (2) 

Solution. 

Multiplying  (2)  by  2,  2xy  =  4.  (5) 

Adding  (5)  and  (1),  x2  +  2  xy  +  2/^  =  9.  (6) 

Subtracting  (5)  from  (1),  x^ -2xy -{•  y^  =  \.  (7) 

From  (6),  x  +  y=±^.  '  '(8) 

From  (7),  x-y  =  ±\,  (9) 

Adding  (8)  and  (9),  2  x  =  ±  3  ±  1.  (10) 

Dividing  (10)  by  2,  and  combining,        a;  =  2,  1,  —  1,  —  2. 
Subtracting  (9)  from  (8),  2y=±^^l,  (11) 

Dividing  (11)  by  2  and  combining,         y  =  1,  2,  —  2,-1. 
Therefore,  the  solutions  of  the  system  (1)  and  (2)  are 

(2,1),  (1,2),  (-1,-2),  (-2,  -1). 

Note.     In  solving    either    the    system    i  ^  ~        \J.\     or 

^  ^  \x  +  y  =  ^       (3)/ 

ra:2  +  2/2  =  5     {1)\ 

\x-y  =  l       (4)/^ 

it  is  evident  that  if  either   (3)    or  ^4)   is   squared  and  combined 
with  (1),  the  value  of  2xy  will  be  obtained;  it  will  be  found  that 


358  ELEMENTARY  ALGEBRA 

2  a;y  =  4,  as  in  equation  (5)  of  the  foregoing  solution  •,  so  that  the 
solution  of  the  system  (1)  and  (3)  reduces  to  the  case  of  the  solution 
of  a  system  such  as  is  given  under  I,  and  the  solution  of  the  system 
(1)  and  (4)  reduces  to  the  case  of  the  solution  of  a  system  such  as  is 
given  under  II.  Observe,  however,  that  the  system  of  equations  (1) 
and  (3),  or  (1)  and  (4)  may  be  solved  by  the  methods  of  section  272. 

1 
EXERCISE    134 

Solve  the  following  systems  : 

^2+^2  =  ¥,  2.     |^  +  ^'  =  4i' 

U  +  y  =  A.  I 


\x-  y  = 


9. 


=  290, 
y  =  16. 
rr2  +  ?/2  ^  13  a2  +  10  a  -h  2, 
X  —  1/  =  a. 

Xy  =  l.  '      \x^=—4:, 

,        o_25a2  +  l6a4-4 
""  ^^   -       (2a+l)2      ' 

^  2a +  1  \xi/=l. 


IV.  When  the  polynomial  in  x  and  y  {obtained  hy  omit- 
ting the  constant  term)  of  one  equation  is  a  factor  of  that  in 
the  other  equation,  a  solution  of  the  system  can  often  be 
obtained  by  quadratic  equations,  even  though  one  of  the 
equations  is  of  a  degree  higher  than  the  second.  The 
solution  of  the  following  system  will  illustrate  : 

ia;8  I   ^8  __    7.  r]\ 

...  , x-^  y  =  \,  (-^) 


SYSTEMS  OF  QUADRATIC  EQUATIONS         359 


Solution. 

Factoring  (1),                (x  +  y)  (x^  -  xy  +  y^)  =  ^. 

(3) 

Substituting  ^ior  x  -\-  yin.  (3), 

J(x^-xy  +  3,«)  =  2V 

(4) 

Simplifying  (4),                            x'' -  xy  +  y^  =  \. 

(5) 

Squaring  (2),                              x^  +  2xy  +  y^  =  \. 

(6) 

Subtracting  (5)  from  (6),                          ^xy  =  —  J. 

(7) 

Dividing,                                                        xy  =  —^. 

(8) 

Subtracting  (8)  from  (5),         x^ -2xy  +  y'^^l. 

(9) 

Therefore,                                                 x  -  y  =  ±\. 

(10) 

Addiug  (10)  and  (2)  and  dividing,               x  =  J  or  —  ^. 

(11) 

Substituting,  in  (2)  the  value  of  x  in  (11),  y  =  -^ov\. 

a2) 

Therefore,  solutions  of   the  given  systems  are   (J,    -^), 

,  and 

(-if)- 

A  solution  similar  to  the  foregoing  may  be  given  for  each 
one  of  the  following  three  systems  of  equations : 

\x-y=ir   \x''-^xy  +  y''=l    ]'  W-xy  +  y''  =  \      I 


EXERCISE    135 

Solve  the  systems : 


=-39, 
13. 


3. 


\oi^  -{-  y^  =  ^.  *U-y  =  l. 

{x+y  =  2,  .    g      {x-y  =  l, 
la;3  4-i/3=98.  •     U3- 2/3  =  127. 

\x^-\-xy  +  y'^=l.  '     Xx^-xy  +  y^^l, 


360  ELEMENTARY  ALGEBRA 


EXERCISE  136.— REVIEW 

Solve  the  following  systems  of  equations 


^^      Sx^--2f  =  l,  ^     \4.x^  +  xy  =  lb. 


7. 


11. 


16. 


17. 


18. 


19. 


rc-f  3«^=34. 

\x^  +  xy  =  Q, 


192^2  +  42/2  =  2, 


—  4  aa;  =  0, 


\6xy  =  l.  ^'    \y  =  mx-{-- 


m 


j9x^-^16y^  =  l,  jy^^Sx, 

1162^  +  9^2  =  1.  »•    \2;2+^2  =  20. 


^'    ^   "     /  =  15.  ^°-    la^-2;y==3«/  +  9. 


jx'- 
\x^- 


4  2^  +  6  2: 2/ +  2  a;  -  6  y  +  1  =  0 , 
2x-{-y-2  =  0. 


12.    [^'  +  3^+11  =  0,  ^^     j2^+^23.74, 


12^2/  = 


\32?  +  2y  +  14  =  0.  12^2/ =  35 

^^      r22:2-2;2/  =  2,  ^^      f2?+32;+y2-2t/=. 

14.      S  ^    o  -<  ^  15'      ' 


12^2_^^=,12.  *"•    \32;-2t/  =  2. 

l22;-3/  = 


a^_^2_  3^.^^  +  2  =  0 
102. 


22^-30^-292^  +  89  =  0, 
y-x  =  2. 

\Sxy  +  2x-Sy-10  =  0, 

\x^-Sxy-^ly^  =  0. 

f22;2-3«/  +  202;-100  =  0, 
[2x^  +  5xy-7y^  =  0, 


20. 


21. 


22. 


23. 


24. 


25. 


28. 


30. 


31. 


SYSTEMS  OF  QUADRATIC  EQUATIONS         361 

'2a;2  + 2a^«/- ^24.  3^_  22/- 15  =  0, 
3a;_4y-2  =  0. 

2a;2  4- 5a;^  -  3  ?/ 4- 2a:  +  36  =  0, 
rr+?/=-2. 

3a^  -  ic?/ +  ^2  _p  22;  -  3^  +  5  =  0, 
3a;2  4.a;^_2i/2  =  o. 

fy2  +  2a:y  +  ^+6  =  0, 
3a:2  4-lla;y  +  6i/2  =  0. 

a;2  _  ^2  _|.  3  a;  _  2  y  4-  237  =  0, 
52:2 -62;?/  + 3/2  =  0. 

f2iry  +  32;-4?/H-19  =  0, 

\l2x-:i/  =  l. 


35. 


22;2- 32:^-23/2  =  3, 
32^2  4.  82:3/ -3^/2  =  8. 

X^-\-  y^z=  a2, 

a;4-^  =  aV2. 


\Q^-xy  +  y'^  =  l, 

X     y  _b 
y     X     b 


27. 


2:8  4. 3/3  =  ^37, 
x  +  y  =  -l. 


f2^4-3/'=«', 
^^-    \2/  =  32;+aVlO. 


32.     ^ 


34. 


J  +  -2=4a2+62 

2^2         ?/2 


xy  = 


2  ah 


36. 


2^*4-2:21/24.3/4^91^ 
2:2  _  0:3/  4-  y2^  7^ 

2:4  _  2/*  =  15^ 


362  ELEMENTARY  ALGEBRA 


37. 


38. 


39. 


f  7a:2  +  34rry  +  39  ?/2  +  a;  +  3^  +  6  =  0, 
1332:2^140^^  +  1473^2=0. 

f92:2_ii2:^_,.  2^2  =  0, 

1 55a;2  -  542:y  -  13  ?/2  +  5a;  +  «/ +  6  =  0. 


EXERCISE  137 

1.  The  sum  of  two  numbers  is  25  ;  the  sum  of  their 
squares  is  457.     Find  the  numbers. 

2.  The  sum  of  the  squares  of  two  numbers  is  225,  and 
the  difference  of  their  squares  is  63.     Find  the  numbers. 

3.  The  ratio  of  two  numbers  is  ^ ;  their  product  is 
315.     Find  the  numbers. 

4.  The  product  of  two  numbers  is  637  and  their 
quotient  is  13.     Find  the  numbers. 

5.  The  product  of  the  sum  and  difference  of  two 
numbers  is  81  and  the  quotient  of  their  sum  divided  by 
their  difference  is  |.     What  are  the  numbers  ? 

6.  Find  two  numbers  whose  sum,  whose  product,  and 
the  difference  of  whose  squares  are  equaL 

7.  The  area  of  a  right  triangle  is  6  sq.  ft. ;  the  hypote- 
nuse is  5  ft.     Find  the  sides. 

8.  The  diagonal  of  a  rectangle  is  25  in.  If  the  rec- 
tangle were  4  in.  shorter  and  8  in.  wider,  the  diagonal 
would  still  be  25  in.     Find  the  area  of  the  rectangle. 

9.  Two  integers  are  in  the  ratio  2:3.  If  each  is 
increased  by  5,  the  difference  of  their  squares  becomes 
40.     What  are  the  integers? 


SYSTEMS  OF  QUADRATIC  EQUATIONS  363 

10.  The  combined  perimeters  of  two  squares  are  68  in. 
One  square  contains  51  sq.  in.  more  than  the  other. 
Find  the  area  of  each. 

11.  The  difference  of  two  numbers  is  5 ;  the  sum  of 
their  reciprocals  is  ^|.       Find  the  numbers. 

12.  The  difference  of  the  cubes  of  two  numbers  is  604 
and  the  sum  of  the  numbers  is  14.     Find  the  numbers. 

13.  The  difference  of  the  terms  of  a  certain  proper 
fraction  is  8  and  the  product  of  this  fraction  by  one  whose 
numerator  and  denominator  exceed  the  numerator  and 
denominator  of  the  given  fraction  by  1  and  5,  respect- 
ively, is  ^.     Find  the  fraction. 

14.  The  diagonals  of  two  rectangles  are  29  ft.  and  5  ft., 
respectively.  The  ratio  of  their  bases  is  7  to  1  and  that 
of  their  altitudes  5  to  1.  What  are  the  dimensions  of 
the  larger  rectangle  ? 

15.  If  the  length  of  a  rectangle  be  increased  by  4  and 
the  breadth  decreased  by  2,  the  area  remains  unchanged ; 
if  the  length  be  decreased  by  4  and  the  breadth  by  2,  the 
area  is  halved.     Find  the  sides  of  the  rectangle. 

16.  The  perimeter  of  a  right  triangle  is  30  ft. ;  its  area 
is  30  sq.  ft.     Find  the  sides  and  the  hypotenuse. 

Suggestion.     Let  x  —  the  number  of  feet  in  the  base. 

Let  y  =  the  number  of  feet  in  the  altitude. 

Then,  x  +  y  +  Vx^  +  y^  =  30,  (1) 

and  xy  =  60.  (2) 

Transposing,  x  -^  y  -  SO  =  -  Vx"^  +  y^.  (3) 

Squaring,  x^  +  y^  +  900  -  QO  x  -  60  y-\-  2  xy  z=  x^  +  y^.  (4) 

Simplifying,  and  substituting  value  of  xy  from  (2), 

x  +  y  =  17.  (5) 


Now  solve  the  system  J  ^     ^     «  * 
"^         [      xy  =  60. 


364  ELEMENTARY  ALGEBRA 

17.  The  perimeter  of  a  right  triangle  is  70  ft.  and  its 
area  is  210  sq.  ft.     Find  the  three  sides  of  the  triangle. 

18.  The  diagonal  of  a  rectangle  is  37  ft.  If  one  side 
were  4  ft.  shorter  and  the  other  2  ft.  longer,  the  area  of  the 
rectangle  would  be  14  sq.  ft.  greater  than  the  area  of  the 
original  rectangle.    Find  the  sides  of  the  original  rectangle. 

19.  A  page  is  to  have  a  margin  at  the  sides  of  ^  in.  and 
one  of  I  in.  at  the  top  and  at  the  bottom ;  it  is  to  con- 
tain 48  sq.  in.  of  printing.  How  large  must  the  page  be 
if  the  length  is  to  exceed  the  width  by  2 J  inches? 

20.  The  fore  wheel  of  a  carriage  makes  six  revolutions 
more  than  the  rear  wheel  in  going  120  yd. ;  if  the  circum- 
ference of  each  wheel  be  increased  one  yard,  the  fore  wheel 
will  make  four  revolutions  more  than  the  rear  wheel  in 
going  the  same  distance.  Find  the  circumference  of  each 
wheel. 

21.  The  circumference  of  the  rear  wheel  of  a  carriage 
is  2  feet  greater  than  the  circumference  of  the  fore  wheel. 
The  fore  wheel  makes  64  more  revolutions  than  the  rear 
wheel  in  traveling  3496  feet.  What  is  the  circumference 
of  each  wheel  ? 

22.  Three  men,  A^  B,  and  (7,  can  do  a  piece  of  work  to- 
gether in  1^  days.  To  do  the  work  alone  A  would  take 
twice  as  long  as  C  and  2  days  longer  than  B.  How  long 
would  it  take  each  to  do  the  work? 

23.  A  rectangular  box  is  8  in.  long.  Its  volume  is 
192  cu.  in.  and  the  area  of  its  six  faces  is  208  sq.  in. 
Find  the  other  two  dimensions  of  this  box. 

24.  The  hypotenuse  of  a  certain  right  triangle  is  10  ft., 
and  its  area  is  24  sq.  ft.  Find  the  base  and  the  altitude 
of  the  triangle. 


CHAPTER  XIV 
PROGRESSIONS 

276.  Series.  A  succession  of  numbers  that  proceed 
according  to  a  fixed  law  is  called  a  series.  The  numbers 
which  form  the  series  are  called  the  terms  of  the  series. 

Thus,  the  sequence  of  numbers  1,  3,  5,  7,  •••  is  a  series  in  which 
the  first  term  is  1,  and  the  second  term  3.  The  law  of  formation  in 
this  series  is  that  any  term  is  obtained  from  the  preceding  by  the 
addition  of  the  number  2. 

277.  Arithmetical  progression.  A  series  in  which  each 
term  is  obtained  from  the  preceding  by  the  addition  of  a 
constant  number  is  called  an  arithmetical  progression. 

Thus,  —  8,  —  4,  0,  4,  8,  12  is  an  arithmetical  progression. 

278.  Common  difference.  The  constant  number  obtained 
by  subtracting  any  term  of  an  arithmetical  progression 
from  the  next  succeeding  term  is  called  the  common  dif- 
ference, or  simply  the  difference. 

Thus,  in  the  arithmetical  progression  2,  5,  8,  11,  the  common  dif- 
ference is  3. 

279.  General  form.  The  general  form  of  an  arithmet- 
ical progression  is  a,  a  -{■  d,  a  +  2  d,  a  +  S  d,  "-  in  which 
a  represents  the  first  term,  and  d  the  difference. 

280.  The  general  term.  By  inspection  it  is  seen  that 
any  term  of  an  arithmetical  progression  is  equal  to  the 
first  term  plus  a  multiple  of  the  difference. 

366 


366  ELEMENTARY  ALGEBRA 

Thus,  in  the  general  form  it  is  obvious  that  the  coefficient  of  rf  in  the 
second  term  is  1,  in  the  third  term  2,  in  the  fourth  term  3.  In  the 
tenth  term  the  coefficient  of  c?  is  9  and  in  the  nth  term  it  is  n  —  1. 
Hence,  if  the  nth  term  of  an  arithmetical  progression  be  denoted  by 
a„,  we  have  the  formula, 

a„  =  a+(n-l)(f.  (1) 

ILLUSTRATIVE  EXAMPLES 

1.  Find  the  fifteenth  term  of  2,  |,  3, ... . 

Solution.     Here  a  =  2,  d  =  ^,  n  =  15. 
Substituting  in  (1),  a^g  =  2  +  (15  -  1)|-,  or  9. 

2.  Write  the  first  three  terms  of  the  series  whose  twelfth 
term  is  6  and  whose  thirty-fifth  term  is  ^-. 

Solution.     Here  ajg  =  6  =  a  +  (12  -  l)rf,  (1) 

and  085  =  -%^-  = « H-  (35  -  l)d.  (2) 

From  (1),  a  +  lie?  =  6.  (3) 

From  (2),  a  +  S4:d  =  \K  (4) 

Subtracting  (3)  from  (4),  23  d  =  S^.  (5) 

Dividing,  d  -  ^'  (6) 

From  (3),  a  =  6  -  11  </  =  -  |.  (7) 

Therefore,  the  required  terms  are  —  J,  0,  |. 

Remark.  Observe  that  when  any  two  terms  of  an  arithmetical 
progression  are  given,  the  first  term  and  the  common  difference  can 
be  found. 

EXERCISE   138 

1.  Find  the  fourth  term  of  1,  2,  3, ...  . 

2.  Find  the  fifth  term  of  2,  -  2,  -  6, ... . 

3.  Find  the  seventh  term  of  5, 8, 11, ... . 

4.  Find  the  fifth  and  sixth  terms  of  —  6,  —  4,  —  2, ... . 

5.  Find    the    fourth    and    fifth    terms   of    a,  a+  b, 
0+2  5,.... 


PROGRESSIONS  367 

6.  Find   the   fourth    and   fifth    terms   of    rr,   2  rr  +  1, 

7.  Find  the  eighth  term  of  2,  —  1,  —  4, ... . 

8.  Find  the  tenth  term  of  2,  7, 12, ...  . 

9.  Find  the  ninth  term  of  J,  |,  1,  ••.  . 

10.  Find  the  tenth  term  of  —  2,  0,  2, ...  . 

11.  Find  the  ninth  term  of  2  a;,  4  a:,  6  a:, ... . 

12.  Find  the  ninth  term  of  «  +  5,  2  a,  3  a  —  6,  ••• . 

13.  Find  the  Tith  term  of  1,  3,  5, ...  . 

14.  Find  the  nth  term  of  —  4,  —  1,  2, ...  . 

15.  Find  the  nth.  term  of  2,  0,  —  2, ...  . 

16.  Find  the  nth  term  oi  2  x,  6  x,S  x, »"  . 

17.  Find  the  nth  term  of  3  a  —  6,  2  a,  a  +  6,  •••  . 

18.  The  third  term  of  an  arithmetical  progression  is  6 
and  the  eighth  term  is  16.  Find  the  first  term  and  the 
common  difference. 

19.  The  ninth  term  of  an  arithmetical  progression  is 
102  and  the  twenty-second  term  is  141.  Find  the  first 
term  and  the  common  difference. 

20.  The  seventh  term  of  an  arithmetical  progression 
is  ^  and  the  twenty -fifth  term  is  ^^-.  Find  the  twelfth 
term. 

21.  The  thirty-first  term  of  an  arithmetical  progression 
is  46  and  the  forty-ninth  term  is  73.  Find  the  common 
difference. 

22.  The  sum  of  the  fifth  and  twenty-fifth  terms  of  an 
arithmetical  progression  is  13,  and  the  forty-ninth  term  is 
15.     Find  the  series. 

23.  The  rth  term  of  the  series  5,  8,  11,  ...is  equal  to 
the  rth  term  of  the  series  61,  57,  53,  ....     Find  r. 


368  ELEMENTARY  ALGEBRA 

« 

281.  Arithmetical  mean.  When  three  numbers  are  in 
arithmetical  progression,  the  second  number  is  called  the 
arithmetical  mean  of  the  other  two. 

Thus  if  a,  x,  c,  are  in  arithmetical  progression,  x  is  the  arithmeti- 
cal mean  of  a  and  c. 

If  a:  is  the  arithmetical  mean  of  a  and  c,  x  may  be  found  in  terms 
of  a  and  c,  thus : 

Since  x  —  a  and  c  —  x  are  each  expressions  for  the  common  differ- 
ence, 

'  X  —  a  =  c  —  X 

therefore,  x  =  ^  ^  ^;  that  is : 

The  arithmetical  mean  of  two  numbers  is  half  their  sum, 

282.  Arithmetical  means.  In  an  arithmetical  progres- 
sion, the  terms  which  stand  between  two  given  terms  are 
called  arithmetical  means  between  the  given  terms. 

ILLUSTRATIVE  EXAMPLE 

Insert  four  arithmetical  means  between  2  and  17. 

Solution.     a»  =  a  +  (n  —  l)c?. 

There  will  be  six  terms  in  all,  of  which  2  is  the  first  term  and  17 
is  the  sixth. 

Hence,  On  =  Oe*  ^hich  is  17. 

Therefore,  17  =  2  +  (6  -  l)d. 

Whence,  c?=  3. 

The  arithmetical  progression  is  2,  5,  8, 11, 14, 17. 

EXERCISE   139 

Find  the  arithmetical  mean  of  : 
1.  12  and  16.     2.    —  4  and  —  12.      3.    a  and  h. 
4.    x-\-  y  and  x—  y.  5.   1  and  a. 

6.    2a-\-b  and  a -{•2  b.  7.  |  and  |. 

8.    -  and  i.  9.    ^—  and  — — 

a  b  x  —  y  x  +  y 


PROGRESSIONS  369 

10.  Insert  three  arithmetical  raeans  between  1  and  ^. 

11.  Insert  three  arithmetical  means  between  230  and 
710. 

12.  Insert  three  arithmetical  means  between  x—  y  and 
x+i/. 

13.  Insert  six  arithmetical  means  between  —  ^  and  J-|. 

283.  Sum  of  an  arithmetical  series.  The  sum  of  n  terms 
of  an  arithmetical  progression  may  be  obtained  as  follows  : 

Representing  the  sum  of  the  first  n  terms  of  the  arithmetical 
progression  by  S^,  we  have 

S„  =  a  +  (a  +  d)  +  (a  +  2d)  -h  -  +  K  -  2d)  +  (a„  -  d)  +  a^. 
or,        Sn  =  an  +  (an  -  d)  -{■  (an  -  2d)  +  '■'  -\-  (a  +  2d)  -\-  (a  -hd)  +  a. 
.-.  25„  =  (a  +  a^)  +  (a  +  a^)  +  (a-^  a„)  +  +  (a  +  a„) 

+  (a  +  an)  +  (a  +  a„) 
=  n(a  +  an). 

.••  Sn  =  ^(a  +  an). 

That  is,  the  formula  for  the  sum  of  the  first  n  terms  of 
an  arithmetical  progression  is 

S„  =  ?(a  +  <^,).  (2) 

Formula  (2)  may  be  expressed  in  words  as  follows : 

The  sum  of  the  first  n  terms  of  an  arithmetical  progression 
is  equal  to  the  arithmetical  mean  of  the  first  and  last  terms 
multiplied  hy  the  number  of  terms. 

By  substituting  in  formula  (2)  the  value  of  a  from 
formula  (1),  section  280,  we  have  a  second  formula  for 
the  sum  of  the  first  n  terms  of  an  arithmetical  progression ; 
namely, 

S»  =  ^[2fl+(n-l)<l.  (8) 


370  ELEMENTARY  ALGEBRA 

Note.  To  find  Sn,  when  n,  a,  and  ««  are  given,  use  formula  (2); 
when  n,  a,  and  d  are  given,  use  formula  (3);  when  a„,  a,  and  d  are 
given,  find  n  by  formula  (1)  and  then  use  formula  (2). 

ILLUSTRATIVE  EXAMPLES 

1.  Find  the  sum  of  the  first  n  natural  numbers. 
Solution.     The  required  sum  =  1  +  2  +  3  +  -. •+n 

_  n(n  +  1) 
2 

2.  Find  the  sum  ofl+3  +  5  +  7+  + 

Solution.     By  formula  (3)  5„  =  |  [2  +  (n  -  1)2]  =  n^. 

Remark.  Observe  that  the  result  of  example  2  expresses  the 
fact  that  the  sum  of  the  first  n  consecutive  odd  numbers  is  a  perfect 
square,  namely,  n^. 

EXERCISE   140 

1.  Find  the  sum  of  15  terms  of  —  1,  4,  9,  ... 

2.  Find  the  sum  of  ten  terms  of  f 9  —  f ^  —  ^,  ••• 

3.  Find  the  sum  of  12  terms  of  3,  ^^,  J5I-,  ... 

4.  Find  the  sum  of  9  terms  of  '^~^,  V3,  ^^"^^  ... 

5.  Find  the  sum  of  n  terms  of  — ^t — ^  —  "^    ,  1  ... 

n  2n 

6.  Find  the  sum  of  m  terms  of  (a  —  1)^,  a^  + 1, 
(«  +  l)2,  ... 

7.  Find  the  sum  of  all  the  even  numbers  between  1 
and  101. 

8.  Find  the  sum  of  all  the  odd  numbers  between  0  and 
100. 

9.  Find  the  sum  of  all  the  multiples  of  5  between  1 
and  1001. 


PROGRESSIONS  371 

10.  Find  the  sum  of  all  the  multiples  of  7  between  1 
and  344. 

11.  The  12th  term  of  an  arithmetical  progression  is  35, 
the  sum  of  the  first  12  terms  is  222.     Find  the  series. 

12.  The  first  term  of  an  arithmetical  progression  is  a,  the 
last  term  is  ?,  and  the  common  difference  is  1.  Show  that 
n  =  Z  —  a  +  1. 

13.  Find  the  ratio  of  the  sum  of  the  first  n  natural 
numbers  to  the  sum  of  the  first  n  even  numbers  beginning 
with  2. 

14.  Show  that  if  each  term  of  an  arithmetical  progres- 
sion be  multiplied  or  divided  by  the  same  number  (zero 
excepted),  the  result  will  be  in  arithmetical  progression. 

15.  A  body  falls  16.08  feet  in  the  first  second,  three 
times  as  far  in  the  next  second,  five  times  as  far  in  the 
third  second,  and  so  on.  How  far  does  it  fall  in  10 
seconds  ? 

284.  Geometric  progression.  A  series  of  numbers  in 
which  the  ratio  of  each  number  to  the  preceding  is  con- 
stant is  called  a  geometric  progression.  The  constant  ratio 
is  called  the  common  ratio  of  the  progression. 

Thus,  the  series  of  numbers,  3,  6, 12,  24,  form  a  geometric  progres- 
sion of  four  terms,  since  J  =  ^^-  —  ^|. 

285.  Increasing  progression.  The  successive  terms  of  a 
geometric  progression  increase  in  absolute  value  if  the 
ratio  is  numerically  greater  than  1,  and  the  progression  is 
said  to  be  an  increasing  geometric  progression. 

Thus  1,  3,  9,  27,  81  is  an  increasing  geometric  progression. 


372  ELEMENTARY  ALGEBRA 

286.  Decreasing  progression.  The  successive  terms  of  a 
geometric  progression  decrease  in  absolute  value  if  the 
ratio  is  numerically  less  than  1,  and  the  progression  is 
said  to  be  a  decreasing  geometric  progression. 

Thus,  128,  64,  32,  16,  8  is  a  decreasing  geometric  progression. 

287.  Form  of  a  geometric  progression.  From  the  defini- 
tion, the  general  form  of  a  geometric  progression  is 

a^  ar^  ar^,  ar^,  ar*,  ••• 
in  which  a  represents  the  first  term  and  r  the  common 
ratio  of  the  progression. 

288.  General  term.  An  inspection  of  the  foregoing 
general  form  shows  that  any  term  of  a  geometric  progres- 
sion is  obtained  by  multiplying  the  first  term  by  a  power 
of  the  ratio. 

Since  the  exponent  of  r  in  the  second  term  is  1,  in  the 
third  term  2,  in  the  fourth  term  3,  it  is  evident  that  in 
the  tenth  term  the  exponent  is  9,  and  in  the  general,  or 
nth.  term,  it  is  ti  —  1.  Hence,  representing  the  nth.  term  by 
an,  we  have  ^^  —  ^^.n-i^  ^1^ 

ILLUSTRATIVE   EXAMPLE 

The  third  term  of  a  geometric  progression  is  ^  and  the 
sixth  term  is  y^^g.     Find  the  series. 

Solution.     Here  a^  =  ar^  =  ^, 

and  a^  =  ar^  =  ^j^. 

S=h'       • 

or,  r8  =  ^^, 

whence,  when  r  is  real,  ^  =  §• 

From  ar'^  =  ^, 

we  have  i^  =  ^j  or  a  =  J. 

Therefore,  the  series  is  -J,  J  x  §,  J  x  (|)^>  •••  *>  that  is,  J,  ^,  ^,  •••. 


PROGRESSIONS  373 

Remark.  Observe  that  when  any  two  terms  of  a  geometric  pro- 
gression are  given,  the  first  term  and  the  common  ratio  can  be  found. 

EXERCISE  141 

1.  Find  the  fifth  term  of  1,  2,  4,  .... 

2.  Find  the  sixth  term  of  3,  |,  |,  ••.. 

3.  Find  the  sixth  term  of  1,  5,  25,  •••. 

4.  Find  the  fifth  term  of  2,  -  4,  8,  .... 

5.  Find  the  sixth  term  of  a,  a6,  a^,  ••-. 

6.  Find  the  fifth  term  of  a,  2a\4:a\  .... 

7.  Find  the  fifth  term  of  m\  m(m  —  V)^  {m—  1)2,  .... 

8.  Find  the  fifth  term  of  1,  V2,  2,  .... 

9.  Find  the  fifth  term  of  ^^~^,  V2,  4(1  +  V2),  .... 

10.  Find  the  nth  term  of  3,  9,  27,  ••.. 

11.  Find  the  nih.  term  of  1,  J,  ^,  •... 

12.  Find  the  n\h.  term  of  a,  Va,  1,  •••. 

13.  The  fourth  term  of  a  geometric  progression  is  24 
and  the  sixth  term  is  96.  Find  the  ratio  and  the  first 
term. 

14.  The  third  term  of  a  geometric  progression  is  16 
and  the  seventh  term  is  Jg.     Find  the  tenth  term. 

15.  The  sum  of  the  first  and  fourth  terms  of  a  geometric 
progression  is  ^Q  and  the  sum  of  the  second  and  third 
terms  is  24.     Find  the  series. 

16.  The  sum  of  three  numbers  in  a  geometric  progres- 
sion is  14  and  the  sum  of  their  squares  is  84.  Find  the 
numbers. 


374  ELEMENTARY  ALGEBRA 

17.  Each  stroke  of  a  certain  air  pump  exhausts  one 
sixteenth  of  the  air  in  the  receiver.  How  much  of  the  air 
originally  in  the  receiver  is  removed  in  six  strokes  ? 


Geometric  mean.  When  three  numbers  are  in 
geometric  progression,  the  second  number  is  called  the 
geometric  mean  of  the  other  two. 

Thus,  if  a,  x,  b  are  in  geometric  progression,  x  is  the  geometric 
mean  of  a  and  h. 

If  X  is  the  geometric  mean  of  a  and  b,  x  may  be  found  in  terms  of 
a  and  b  thus : 

Since  -  =  r  and  -  =  r, 

a  X 

we  have  -  =  -. 

a     X 

Therefore,  x^  =  a&, 

and  X  =  Va6  ;  that  is : 

77ie  geometric  mean  of  two  numbers  is  the  square  root  of 
their  product. 

Remark.  It  is  evident  that  the  geometric  mean  of  two  numbers 
is  the  mean  proportional  between  these  numbers  [§  175]. 

290.  Geometric  means.  The  terms  which  stand  be- 
tween any  two  given  terms  of  a  geometric  progression 
are  called  the  geometric  means  between  the  given  terms. 

Thus,  in  the  geometric  progression  2,  4,  8,  16,  32,  the  geometric 
means  between  2  and  32  are  4,  8,  and  16. 

ILLUSTRATIVE  EXAMPLE 

Insert  three  real  geometric  means  between  32  and  2. 
Solution.  a„  =  ar'^'^. 

There  are  five  terms  in  all,  of  which  32  is  the  first  term  and  2  the 
fifth  term. 

Here  a„  =  Og  =  2  =  ar^. 

.•.82r*  =  2. 

Whence^  '^  =  :^'  ^^^  '*  =  ± 5  ^^^  ± \' 

lo  «  « 


PROGRESSIONS  375 

That  is,  the  two  real  values  of  r  are  |  and  —  4. 
Corresponding  to  the  real  values  of  r,  we  have  the  progressions 

32,  16,  8,  4,  2,  and  32,  -  16,  8,  -  4,  2. 
The  required  means  are,  therefore,  either 

16,  8,  and  4  or  -  16,  8,  and  -  4. 

EXERCISE  142 

Find  the  positive  geometric  mean  between : 
1.    3  and  48.  2.    Jg  and  J^. 

3.    ^  and  -.  4.    (x-yy  and  (x^-^y^y, 

5.    3  a  and  27  a^.  6.    a  and  a^. 

Insert  three  positive  geometric  means  between : 
7.    4  and  64.  8.    48  and  243. 

9.    (x  —  y')  and  (^x^  —  y^^(x  -\-  y')^, 

10.  -  and  — . 

0  a^ 

11.  The  sum  of  three  numbers  in  geometric  progression 
is  117 ;  the  mean  is  equal  to  three  tenths  of  the  sum  of 
the  other  numbers.     Find  the  numbers. 

291.  Sum  of  a  geometric  series.  Let  S^  represent  the 
sum  of  n  terms  of  a  geometric  progression ;  then, 

5„  =  a  +  ar  +  ar2  +  ...  4-  ar""-^  +  ar""-^  +  ar''-\  (1) 

Multiplying  (1)  by  r, 

r  5„  =         ar  -\-  ar^  +  •••  +  ar'^'^  +  ar^^  +  ar^,  (2) 

Subtracting  (2)  from  (1), 
(1  -  r)5„  =  a  -  ar"". 
Dividing,  , 

o„=  —1 


376  ELEMENTARY  ALGEBRA 

That  is,  the  formula  for  the  sum  of  n  terms  of  a  geo- 
metric progression  is 

S„  =  ^Z^.  (2) 

In  an  increasing  geometric  progression  the  formula 
obtained  by  changing  the  signs  of  the  terms  in  formula 
(2)  should  be  employed ;  the  formula  is, 

S„  =  ^«.  (3) 

ILLUSTRATIVE  EXAMPLES 

1.  Find  the  sum  of  eight  terms  of  2,  6, 18,  •••. 
Solution.     Here  a  =  2,   r  =  S,   n  =  8. 

We  use  the  formula  Sn  = — —  • 

r  —  1 

Substituting  in  the  formula, 

S.  =  ^'^^-^  =  38  -  1  =  6560. 
8  2 

2.  The  sum  of  the  terms  of  a  geometric  progression  is 
728,  the  ratio  is  3,  and  the  last  term  is  486.  Find  the  first 
term  and  the  number  of  terms. 


Solution.     Since 

«n  =  ar»~\ 

ran  =  o^", 

and  the  formula 

s„  =  '""°-" 

r-1 

may  be  written, 

-.=?fr- 

Here,  5„  =  728,  r  = 

=  3,  and  a„  =  486. 

Substituting, 

^„o      3  X  486  -  a 
^^^-       3-1 

whence, 

a  =  2. 

From 

a„  =  ar«-i. 

we  have 

486  =  2  X  S»-\ 

or. 

3«  =  3«-i. 

Hence, 

5  =  n  -  1, 

or, 

n  =  6. 

PROGRESSIONS  377 

EXERCISE  143 

1.  Find  the  sum  of  2,  6, 18,  •-.  to  6  terms. 

2.  Find  the  sum  of  4,  2,  1,  ...  to  8  terms. 

3.  Find  the  sum  of  —  3,  9,  —  27,  ••.  to  7  terms. 

4.  Find  the  sum  of   V3,   3  +  V3,  6  +  4V3,  ...  to  5 
terms. 

5.  Find  the  sum  of  1,  2,  4,  •••  to  7i  terms. 

6.  Find  the  sum  of  1,  J,  ^,  •••  to  ti  terms. 

7.  Find  the  sum  of  aV5,  6Va,  5V5,  ...  to  n  terms. 

8.  Find  the  sum  of  3,  —  6, 12,  ...  to  2  m  +  1  terms. 

9.  If  the  sum  of  the  series  1  -|-  4  +  16  +  •••  is  5461, 
find  the  number  of  terms. 

10.  The  first  term  of  a  geometric  progression  is  1,  the 
last  term  is  81,  and  the  sum  of  the  series  is  121.  Find 
the  ratio. 

11.  Find  two  numbers  whose  sum  is  52,  such  that  their 
arithmetical  mean  exceeds  their  geometric  mean  by  2. 

12.  The  first  term  of  a  geometric  progression  is  5,  the 
ratio  is  4,  and  the  number  of  terms  is  5.  Find  the  sum 
of  the  terms. 

13.  The  first  four  terms  of  a  geometric  progression  are 
the  same  as  the  first  four  terms  of  a  second  geometric 
progression,  but  in  reverse  order  ;  the  sum  of  the  first 
eight  terms  of  one  is  equal  to  81  times  the  sum  of  the 
first  eight  terms  of  the  other.  Find  the  common  ratio  of 
each. 

292.  Infinite  geometric  series.  The  terms  of  a  geometric 
progression  in  which  r  is  positive  and  numerically  less 
than    1,   become    smaller   and   smaller.      By   taking    n 


378  ELEMENTARY  ALGEBRA 

sufficiently  large,  we  can  make  the  nth  term  as  small  as 
we  desire  ;  that  is,  make  it  more  and  more  nearly  equal 
to  zero. 

By  this  statement  we  mean  that  however  small  a'  num- 
ber we  please  to  mention,  we  can  find  a  term  such  that 
it  and  each  one  of  the  succeeding  terms  of  the  series  is 
numerically  less  than  the  number  mentioned. 

Thus,  in  the  series  100,  10,  1,  J^,  yoo"'  T"^Vo'  *"  *^®  successive 
terms  are  becoming  smaller  and  smaller.  When  n  is  10,  a„  is  y^nToV^nrS"' 

which  is  a  small  number;  and  when  n  is  103,  a„  = ,  which  is  an 

exceedingly  small  number. 

From  illustrative  example  2,  section  291, 
o  _a  —  ra^ 


which  may  be  written,  aS'^  = aJ ). 

1  —  r  \1  —  rj 

The  term is  a  constant,  and  the  term  aJ — - — ) 

1  —  r  \1  —  rJ 

varies  with  n  and  becomes  smaller  and  smaller.     Hence, 

as  more  and  more  terms  of  the  series  are  added,  aS^^  differs 

less  and  less  from . 

1  —  r 

The  number is  called  the  limit  of  the  sum  of  n 

1  —  r 

terms,  as  n  increases  without  limit. 

For  convenience  we  shall  call  this   limit   the   sum  to 

infinity  of  a  decreasing  geometric  progression,  and  shall 

denote  it  by  the  symbol  S^  .      We   therefore   have  the 

following  identity  : 

<3     -      ^ 


PROGRESSIONS  379 

293.  Recurring  decimal.  A  decimal  in  which  a  figure 
or  set  of  figures  repeats  in  a  certain  fixed  order  is  called  a 
recurring  decimal,  or  a  repeating,  or  circulating  decimal. 

Recurring  decimals  are  illustrations  of  infinite  geometric  series : 
Thus,  each  of  the  following  is  an  infinite  geometric  series : 

.6666  •••,  sometimes  written  .6,  in  which  a  =  .6  and  r  —  .1. 

.3434  •..,  sometimes  written  .34,  in  which  a  =  .34  and  r  —  .01. 

.304304  •..,  sometimes  written .304,  in  which  a  -  .304  and  r  -  .001. 

ILLUSTRATIVE  EXAMPLES 

1.  Sum  to  infinity  the  series  2, 1,  |,  ;^,  ••• 
Solution.  Here  a  =  2  and  r  =  |^. 

2.  Sum  to  infinity  the  series  6,  —  3, 1|,  —  |  •••. 
Solution.  Here  a  =  6  and  r  =  —  -^. 

S    -     Q     -         ^         -_L-4 

3.  Sum  to  infinity  the  series  a, —^ — , ^ ,  .... 

Solution.  Here  a  =  a,  and  r  =  — 

a;2  +  l 


^-^      1- 


a:2  +  l 

4.  Sum  to  infinity  the  series  .666  •  •  • . 
Solution.  Here  a  =  .6  and  r  =  .1. 

S    =     ^     =     -^     -'^-2 
"      l-r      1-.1".9~3' 

5.  Sum  to  infinity  the  series  .24545  •••. 

Solution.    .24545  •••  =  .2  +  .04545  •••,  in  which  .2  is  not  a  part  of 
the  series.  jjere  a  =  .045  and  r  =  .01. 

S    ^     q     ^    .045    ^.045^    5 
*      l-r     1  -  .01       .99      no' 

•••  •2^5  =  1^  +  ^4^,  or  ^ViJ. 


380  ELEMENTARY  ALGEBRA 

,       EXERCISE  144 

1.  Sum  to  infinity  the  series  1,  ^,  J,  .••. 

2.  Sum  to  infinity  the  series  100, 10, 1,  •••. 

3.  Sum  to  infinity  the  series  5, 1,  ^,  •••. 

4.  Sum  to  infinity  the  series  1,  J,  ^,  •••. 

5.  Sum  to  infinity  the  series  6,  |,  -^^^  •*•• 

6.  Sum  to  infinity  the  series  |,  |,  f,  •••• 

7.  Sum  to  infinity  the  series  2.5, 1.25,  .625,  •••. 

8.  Sum  to  infinity  the  series  18, 12,  8,  .•-. 

9.  Si;m  to  infinity  the  series  3.5,  .35,  .035,  •••. 

Sum  to  infinity  the  following : 

10.    .5.  11.    .54.  12.   .61. 

13.    .i35.  14.    .24.  15.    .434. 

16.  Show  that  4  +  1  +  ^/  +  —  =3  +  f +  |J  +  •  — 

17.  An  elastic  ball  bounces  to  three  fourths  the  height 
from  which  it  falls.  If  it  is  thrown  up  from  the  ground 
to  a  height  of  20  feet,  find  the  total  distance  traveled 
before  it  comes  to  rest. 

18.  A  heavy  iron  ball  at  the  end  of  a  chain  is  pulled 
to  the  right  1  yard  out  of  the  vertical  and  is  then  released. 
It  swings  to  a  point  0.9  of  a  yard  to  the  left  of  the  vertical, 
then  to  a  point  0.9  of  a  yard  to  the  right  of  the  vertical. 
The  succeeding  swings  follow  the  same  law.  Including 
the  first  movement,  find  the  greatest  distance  the  ball 
could  travel  before  coming  to  rest. 

19.  Show  that  before  the  ball  mentioned  in  example  18 
passes  through  the  vertical  for  the  seventh  time  after 
being  withdrawn,  it  has  moved  more  than  half  its  total 
movement. 


CHAPTER  XV 
GENERAL  REVIEW 

1.  Evaluate 1 -—   when  a  =  3,  5  =  4,  and 

c=l. 

2.  Evaluate  — — — — ±- i—  when  x  =  2. 

3.  Evaluate  ^P  "  ?)'  +  C  +  *) %  (P  +  ?)'  +  C"  -  O'  ^ 

p  -{■  q  +  r  -\-  8  p  —  q-\-  r  —  8 

when  JO  =  3,  3'  =  2,  r  =  1,  and  8  =  ^ . 

4.  Write  a  formula  for  the  area  of  a  parallelogram. 
Find  the  area  of  a  parallelogram  whose  base  is  10  inches 
and  whose  altitude  is  7  inches. 

5.  Write  a  formula  for  each  of  two  numbers  whose  sum 
and  difference  are  known.  Find  two  numbers  whose  sum 
is  126  and  whose  difference  is  32. 

6.  Write  a  formula  to  find  the  weight  of  a  bag  con- 
taining any  given  number  (n)  of  bushels  of  grain,  given 
the  weight  of  the  bag  and  the  weight  of  a  single  bushel 
of  grain. 

7.  Write  a  formula  for  the  area  of  the  wall  of  a  room 
of  length  L  and  width  TF"  containing  two  windows  each 
of  length  I  and  width  w, 

8.  Write  a  formula  to  find  the  number  of  square  feet 
in  the  four  walls  of  a  room,  I  ft.  long,  w  ft.  wide,  and  h  ft. 
high. 

381 


382  ELEMENTARY  ALGEBRA 

Find  the  sum  of  the  expressions  in  examples  9-12. 

9.    Sa^-\-2xt/-4f  +  lSz^,  2x^-5x1/-^  7t/^-^Uz^ 
and  -3a;2+  2x^-7 y^-5z\ 

10.  8  mn^  —  7  mhi  -{-  2n^,  5m^  —  2  mhi  —  11  mn\  and 
m^  —  2  nK 

11.  (jE?  -h  q)a,  (^q  +  r)a,  and  (r  —  5')a. 

12.  ix-^^-\-^z,^x  +  ^y-^z,and2x-ii/-^^z, 

13.  Simplify  3(a  -  b -\- c -\-2d}- 5(a- 2b+ Sc--Sd^ 
+  4(a-36  +  2(?  +  4(^)-2(a-7  6-2c-7c^). 

14.  Evaluate  (3  a:  +  2  a)2  -  (2  a;  +  5  ay  when  a;  =  7  a. 

15.  Evaluate  5(2  m  -  37i)(3m  -  2  w)  -  (2/1  +  3  w) 
(3  w  +  2  w)  when  w  =  3  /t. 

16.  Simplify  2a; -(- 3^  +  2- jrr- y|)-(3a;H- 22- 
[-2^  +  32]). 

17.  Simplify       _  5  _  [_  (_  «  4.  J  _  c)]_  cj  -  { -  [- 

18.  From  the  sum  of  3  a6  —  2  a;y  +  4  and  2  aft  —  3  a;^  +  3 
take  the  difference  between  4  a6  +  3  a;y  —  2  and  5  ah  — 2  xy. 

19.  Add  5jo  -  (3  ^-  2  r)  and  -  (3  ^  -6 j^)-  lOjt? ;  from 
the  sum  subtract  —  4^  —  (3  r  +  5'). 

20.  State  what  value  of  x  will  make  the  expression 
3(a;  H-  3)  —  2(2  a:  —  3)  equal  to  twice  the  value  of  x, 

21.  What  number  is  as  much  greater  than  30  as  it  is 
less  than  74  ? 

22.  Find  a  number  to  which  if  10  be  added  the  result 
is  equal  to  6  times  the  number. 

23.  A  man,  who  rode  a  motorcycle  at  the  rate  of  m 
miles  an  hour,  completed  a  journey  from  P  to  ^  in  A 
hours,  during  r  of  which  he  rested.  Find  an  expression 
for  the  distance  from  P  to  ©• 


GENERAL  REVIEW  383 

24.  Multiply  5a;3-22^  +  7a;-llby3ar»  +  7a;-3. 

25.  Simplify  (ax  +  hy)(cx  —  dy)  —  (ex  —  hy)(ax-\-  dy), 

26.  Multiply  b(x  +  yy  -  3(a;  +  ^)  -  2  by  3(a:  -f-  y)- 

27.  Multiply  .3  2^2  ^1,2  a; +1  by  .52^2  _i. 

28.  Multiply  a;"* +  2  2^-1  + 3  a^-2  by  2:1; -3. 

29.  Multiply  a°+i  —  2  a«  +  3  by  2  a"  —  3.     Verify  your 
result  by  putting  a  =  2. 

30.  Simplify  ba-\'^a-[2h(p -^  q)-^h(p  -  q)^\, 

31.  Prove  by  actual  multiplication  that 

(^a  +  h+cy-\-a^  +  y^+(^=(h+cf+(c  +  ay+(a^hy. 

32.  Show  that  if  2;  =  1  +  a,   y  =  1  +  5,   z  =  l-\-  c,  then 
a;2  -(-  ^2  ^  2;2  —  ^2  —  237  —  2;^  =  a^  _|_  52  _|_  ^  _  5^  __  ^^  _  ^5^ 

33.  Prove  by  actual  multiplication  that 
(a^  —  bey—  (h^  —  ed)(c^  —  aJ))  is  equal  to 

a(a  +  h-\-  c)(a2  +  5^  +  c^  —  6c  —  ca  —  ah), 

34.  Arrange       the      expression      a^(x  —  1)  -j-  (oc^  —  1)^ 
—  mx  (a^  -f  3)  in  descending  powers  of  x. 

35.  Divide4a3  4-a^-i3by  a-|. 

36.  Divide  a^^  +  53^  by  a^  +  ^. 

37.  Divide  a;^  +  8  a;^  -  192  a:^  _  256  x  +  1024  by  a:^  _  4  ^^s 
-16a;+32. 

38.  Divide  (x-\-6  yY  —  (y -^  4:  zy  hj  x  •{-  4:(y  —  2). 

39.  Divide  6x^ -9x^ +  22a^- 4:0^ --^x-lSa^-^lOhy 
3a:3_2a:  +  5. 

40.  Divide    a^+"6"  —  5  ^r+^-^J^n  _  3  ^-|-n-258n 
+  15  a"»+»-354n  lyj  fjfiln  _  5  ^n-152n^ 

41.  Factor24-822_9. 

42.  Factor  (m  +  2)*  -  9(w  +  2)2+  20. 

43.  Factor  a^  —  b^. 


384  ELEMENTAKY  ALGEBRA 

44.  Factor  aa^  -\- bx^ -{- a  —  h. 

45.  Factor  a^ -\- 2  ab -\- S  ac -\- 6  he. 

46.  Factor  a^-b^-  a{a^  -  ^2)  +  6(a  -  b)\ 

47.  Factor  {i^  +  3  2^)2  -  (3  x^  +  l)^. 

48.  Factor  ^^-|i. 

«>6        27 

49.  Factor^2;2_|.2^a;- 3^  +  ^2^^  +  2^x-3^. 

50.  Factor  3(a:-3)  +  a;(rc-3)(3a;+3). 

51.  Factor  a!^  -2aW  ^b^  -  a^  -\-2  ab  ^  W^. 

52.  Prove  that  {tP'  -\- xi/ 4-  y^)^-  (aP-^XT/-  y^y 
=  ^xy\x-^y), 

53.  Factor  j^^  _  3  ^2  ^.  4, 

54.  Factor  7^  —  x^y  —  xy'^  +  y^, 

55.  Factor  jt?2  _  6  j9^- 16^252 +  9^. 

56.  Factor  (x^  -lxy-\-  18(a^  -  7  a;)  +  72. 

57.  Factor  x^^  +  m^xy^. 

58.  Factor  3(2^+1)84-4(2;  + 1)2  + a; +  1. 

59.  Express  (5  a;  -  6)  (5  a;  +  6)  -  4  ^(10  a;  -  4  ^)  as  the 
difference  of  two  squares. 

60.  Resolve  a:^  —  13  a;2  +  36  into  linear  factors. 

61.  Show  that 

(9  a:  ~  10)2-  2(7  X  -  10)2=  (x  -  10)2  _  2(3  a;  - 10)3. 

62.  Find  the  H.C.F.  and  L.C.M.  of  a^s+l  and  2  a:2_a;_ 3. 

63.  Find  theH.C.F.  and  L.C.M.  of  a:3  +  22^  +  2a;+l 
anda:8-.2a;2  +  2a;-l. 

64.  Reduce  to  lowest  terms:   '^'~^'  +  f+^^^ 

a^  —  c^— 0^  —  2  bo 

TO       ''  2 

65.  Simplify      -j f—J' 


.  GENERAL  EEVIEW  385 

66.    Simplify 

x-hB  ,  x  +  2  ,  a; +3 


(2_^)(3-jr)      (x-^X^-^}      (a;-2)(a:-5) 
67.    Simplify 


+ 


(a  —  6)  (b  —  e)      (h  —  c)(^c—  a)      (a  —  c)(h—  a) 
69.    Simplify 

\x—y     x^-  y     aP'-y'^]    '\x  +  y     x^y] 


71.  Simplify     fsx-'5--Ysx-hB--\-^(x--\ 

72.  Simplify 


(a— 6)(a--c)      (6  — c)(6  — a)      (c— a)(<?  — 6) 


2:3  +  27 


73.  Simplify     —^±^—-2^1 ?-V 

„,     o-      r-P         2        1       2a  +  3,l,3«-26 

74.  Simplify     —---___— n_+       4- _^ — 

3a      26        Qa^        2  a         o  ao 

o-      Tx         abc—a%        ahc—h^c        ahc  —  c^a 

75.  Simplify     -— — -  X  —rz —  X 


a%e  —  ^6^^     a62c  —  abc^     ahc^  —  a26c 

76.  Simplify    [m^^     f     V^2^^2)^/^_^_  +  _JL-Y 

V         m^—n^J  \m+n     m—nj 

77.  Evaluate     aa?  +  6v  +  c    when    a^  = — — -    and 

ma  —  po 
_  pc  —  na  ^ 

ph  —  ma 


386 


ELEMENTARY  ALGEBRA 

1  r  4 


x-^- 


78,  Simplify 

79.  Simplify- 
so.    Simplify 


x-^ 


X—  5 


x  —  b 


a;-3 


X—K>   — 


x  —  b 


a^  —  ah      b^—bc      c^—  ca 
c^  —  ac     b^—ba     c^ _  ^^ 

^_^_2i(a-ft) 
a-\-b 
a2  +  ^2 


a6  +  ^2 


81.    Show  that  ^^-f^  +  ^^±^  =  (^  +  ^)^^±-^. 
a+6  a-b        ^  ^  a^-b^ 


82. 


Simplify     f-l^ L  +  _8^V-^^=^ 

V2a;+^      2x-y^  y'^-4.x^J     (Ix-y) 


(2x-yy 


83.    Solve  the  equation     ^^±2  ^  ^±1  ^  2. 


84.  Solve  the  equation 

85.  Solve  the  equation 

86.  Solve  the  equation 

87.  Solve  the  system 


3 


5  +  4a;         2x 


3a;-4      Qx-1     3a:-4 
m     x     X     n     p 

^X  X 


x-\-2     x+1 


=  2. 


x-^y     x-y_  -. 
3  4     "      * 

~2 3     -^- 


88.  Solve  the  system 

89.  Solve  graphically  the  system 


2xy-\-Sx  =  6, 
Sxy  +  6x  =  S. 

'2x-y=:i, 


2x+Sy==12. 


GENERAL  REVIEW 


387 


90.  Divide  m  into  two  parts,  one  of  which  shall  exceed 
the  other  by  n, 

91.  The  difference  of  the  squares  of  two  consecutive 
odd  numbers  is  96.     Find  the  numbers. 

5 


92.    Solve  the  system 


by 


=  -2,    x+ly  =  ^. 


ft3.    Solve  the  system 

94.    Solve  the  system 
3       .       5 


=  8, 


=  11, 


x-2     y+3 
2  3 


=  11. 


'2,x  +  y     x  —  ^ 


2x-\-  y     X—  Sy 

95.  Solve  the  system 

f4a;H-8y-32  =  6, 
hx-{-  y-  2  =  7, 
[4^  — 5  a; +  4  2  =  8. 

96.  Solve  the  system 

^_3        y     _^        ^_^ 

3+^~7'    4  +  2"9'   5  +  x~8' 

97.  Solve  the  system 
5      .^     4 

x      y      z 
§-2  +  1=12. 

98.  Solve  the  system 
4  5 

5£:iI  +  ii^z:^  =  i8-5  3,. 


=  1. 


388  ELEMENTARY  ALGEBRA 

99.    Solve  for  a;  and  ^:   -^  +  X.  =  ^,   -  +  ^=2. 

oa     2  0     6     a      b 

100.  A  mixture  of  corn  and  oats  contains  33J  %  of  oats 
by  weight.  How  many  pounds  of  corn  must  be  added  to 
100  lb.  of  the  mixture  so  that  the  resulting  mixture  shall 
contain  only  20  %  of  oats? 

101.  A  dealer  bought  2000  lemons,  some  of  them  at  the 
rate  of  1 J  ct.  apiece,  and  the  remainder  at  the  rate  of  2  ct. 
apiece.  He  sold  them  all  at  the  rate  of  27  ct.  per  dozen 
and  gained  $7.50.     How  many  did  he  buy  at  each  price? 

102.  If  7  is  added  to  twice  a  certain  number,  the  sum  is 
13.     Find  the  number. 

103.  If  one  half  of  a  certain  number  is  added  to  itself, 
the  sum  is  3  less  than  twice  the  number.  Find  the 
number. 

104.  Twice  a  certain  number  is  9  less  than  5  times  the 
number.     What  is  the  number? 

105.  One  number  is  3  times  a  second  number ;  the  sum 
of  the  two  numbers  is  6  greater  than  twice  the  smaller 
number.     What  are  the  numbers  ? 

106.  The  sum  of  two  numbers  is  50 ;  one  of  the  num- 
bers is  5  less  than  4  times  the  other.  What  are  the 
numbers  ? 

107.  The  difference  between  two  numbers  is  37.  The 
smaller  number  plus  3  times  the  larger  equals  163.  Find 
the  numbers. 

108.  The  sum  of  two  numbers  is  60 ;  one  number  is  17 
less  than  6  times  the  other.     Find  the  numbers. 

109.  One  number  exceeds  another  by  30  ;  the  smaller 
is  3  greater  than  one  half  of  the  larger.  Find  the 
numbers. 


GENERAL  REVIEW  ^89 

110.  One  number  exceeds  another  by  101  ;  if  3  times 
the  smaller  is  added  to  the  greater,  the  result  is  201. 
Find  the  numbers. 

111.  A's  share  of  a  business  is  twice  that  of  his  partner 
B;  they  sell  the  business  for  $12,000.  How  much  should 
each  receive  ? 

112.  The  sum  of  two  numbers  is  280,  and  their  difference 
is  equal  to  one  fourth  of  the  greater.     Find  the  numbers. 

113.  A  house  and  a  garage  cost  $7000,  and  twice  the 
cost  of  the  house  was  equal  to  five  times  the  cost  of  the 
garage.     Find  the  cost  of  each. 

114.  A  number  is  composed  of  two  digits ;  the  digit  in 
the  tens'  place  is  one  less  than  twice  that  in  the  units' 
place.  If  27  is  subtracted  from  the  number,  the  remain- 
der is  composed  of  the  same  two  digits  in  reversed  order. 
Find  the  number. 

115.  The  difference  between  the  squares  of  two  con- 
secutive numbers  is  13.     Find  the  numbers. 

116.  If  A  can  perform  a  piece  of  work  in  3  days,  and  B 
in  5  days,  in  what  time  should  they  perform  it  working 
together  ? 

117.  If  a  man  and  2  boys  can  do  a  piece  of  work  in 
5  days  and  the  man  working  alone  can  do  it  in  12  days, 
in  what  time  can  one  boy  working  alone  do  the  work,  pro- 
viding the  boys  do  equal  amounts  ? 

118.  The  sum  of  the  two  digits  of  a  number  is  14 ;  if  the 
order  of  the  digits  is  reversed,  the  number  is  diminished 
by  18.     Find  the  number. 

119.  A  person  has  just  ten  hours  at  his  disposal ;  how 
far  may  he  ride  at  the  rate  of  ten  miles  an  hour,  so  as  to 
return  home  on  time,  walking  back  at  the  rate  of  4  miles 
an  hour? 


39a  ELEMENTARY  ALGEBRA 

120.  A  train  travels  from  Philadelphia  to  New  York  in 
2  hours  ;  if  it  had  traveled  15  miles  an  hour  slower,  it 
would  have  taken  one  hour  longer.  Find  the  distance 
from  Philadelphia  to  New  York. 

121.  A  number  of  workmen,  who  receive  the  same 
wages,  earn  together  a  certain  sum.  Had  there  been  6 
more  workmen,  and  had  each  received  10  cents  more,  their 
joint  earnings  would  have  increased  by  $19.60.  Had 
there  been  3  fewer  workmen  and  had  each  received  10 
cents  less,  their  joint  earnings  would  have  decreased  by 
$9.70.  How  many  workmen  are  there  and  how  much 
does  each  receive? 

122.  The  total  number  of  boys  and  girls  attending  a 
certain  boarding  school  is  95.  If  the.  number  of  boys 
were  30  %  less  and  the  number  of  girls  20  %  more,  there 
would  be  as  many  girls  in  attendance  as  boys.  How 
many  of  each  are  there  in  attendance? 

123.  A  man  has  $2.20  in  nickels,  dimes,  and  quarters, 
15  coins  in  all.  If  the  number  of  nickels  and  quarters 
were  interchanged,  he  would  have  $1.80.  How  many  of 
each  has  he  ? 

124.  A  quantity  of  wheat  sufficient  to  fill  three  bins  of 
different  sizes  will  fill  the  smallest  bin  four  times,  the 
second  bin  three  times,  or  the  largest  bin  twice  with  40  bu. 
to  spare.     What  is  the  capacity  of  each  bin  ? 

125.  A  wholesale  egg  dealer  sold  on  the  average  3800 
dozen  eggs  a  day  for  cash.  He  reduced  his  price  5%, 
and  found  that  his  average  daily  cash  receipts  from  sales 
were  increased  10%.  How  many  dozen  eggs  did  he  sell 
daily  at  the  reduced  prices? 

126.  A  man  has  $5000  which  he  wishes  to  invest  in 
two  enterprises  so  that  his  total  income  will  be  $  180 ;  if 


GENERAL  REVIEW  391 

one   enterprise   pays  4%  and  the  other  3%,  how  much 
must  he  invest  in  each? 

127.  The  circumference  of  the  rear  wheel  of  a  carriage 
is  3J  ft.  greater  than  the  circumference  of  the  front 
wheel.  The  front  wheel  makes  98  more  revolutions  than 
the  rear  wheel  in  traveling  5600  ft.  What  is  the  circum- 
ference of  each  wheel? 

128.  The  sum  of  two  fractions  is  ^  and  their  difference 
is  y2_.     What  are  the  fractions  ? 

129.  Find  a  fourth  proportional  to  a*,  ah\  6  a%, 

130.  Find  a  mean  proportional  between  32  aV  and  2  a^x. 

131.  Two  numbers  are  in  the  ratio  oi  mi  n.  If  c  be 
added  to  the  first  and  subtracted  from  the  second,  the 
results  will  be  in  the  ratio  of  4  :  5.     Find  the  numbers. 

132.  What  number  must  be  subtracted  from  each  of  the 
numbers  6,  9,  15,  and  27,  so  that  the  resulting  differences 
shall  form  a  proportion  when  taken  in  the  given  order? 

133.  If =  - ,  prove  that  6  is  a  mean  proportional 

b  —  c      c 

between  a  and  c. 

134.  Two  numbers  have  the  ratio  of  7:8;  if  21  be 
added  to  each,  they  have  the  ratio  of  10  :  11.  Find  the 
numbers. 

135.  Represent  by  a  graph  the  distance  traveled  by  an 
automobile  at  the  rate  of  25  miles  an  hour.  What  is  the 
equation  connecting  the  distance  and  the  time? 

136.  Determine    the    value   of   x  from  the  proportion 

137.  If  h  is  a  mean  proportional  between  a  and  <?,  prove 
that  a-26:6-2c  =  2a-35:25-3c. 


392  ELEMENTARY  ALGEBRA 

138.  Two  numbers,  x  and  y  (the  first  being  negative) 
are  in  the  ratio  of  7:  —  10.  If  15  be  subtracted  from  each 
one,  the  resulting  numbers  are  in  the  ratio  of  —10:7. 
Find  the  numbers. 

139.  If  x-^-yi  a-\-h  =  x— y:a—  h^  show  that  x  varies 
as  «/.  . 

140.  Expand  (2-f  a^)*. 

141.  Expand  ia^^ J    by  the  binomial  theorem. 

142.  Find  the  numerical  value  of '- 

2-2 .  27^  •  50 

143.  Express  in  simplest  form  with  positive  exponents: 

24  x~'^yH~^ 
40  x-'^yH-^ 

144.  Find  one  side  of  a  square  whose  area  is  repre- 
sented by  the  following  expression: 

^  _  a:  +  4  -  -  +  -^.    Check  the  result. 
4  ic      ar 

x^yjx~^y^ 

"^yz^ 

146.  Simplify        ^  ^"^^      -^  a^iH. 

at(a  -  ^)o 

147.  If  a  =  64,  express  each  of  the  following  as  an  in- 
teger or  a  fraction:  a^;  -^a"^;  (o^)"^;   [(\^a)^]~^. 

,.«     c-      ^'f       4tx-^y-^     6ic2a-i 

148.  Simplify -^  X 1-:- 

149.  If  a  +  b  +  c=28  and  a  =15,  h=  14,  c=13,  find 
the  value  of  V«(«  —  a)(«  —  6)(«  —  c). 


145.    Simplify 


GENERAL  REVIEW  393 


150.    Simplify     ahH^Y  ^  C^h-^^-^. 


151.  Simplify     ( V«*)  ^• 

152.  Simplify     (25-3 -- a-^rc-S)"*. 


153.  Evaluate   2^  x  9^  x  Srl 

154.  Simplify,  using  positive  exponents  to  express  the 


answer,  m  Vn  % 

155.  Expand  and  express  in  simplest  form  with  positive 
exponents     (m~^a~^  —  ma^^^. 

156.  Simplify    VIOOO;  </ST;  V{^;  -K;  (27*)^;  2^  •  4^ 

■  V2 

m+n  m—n  2to— 1 

157.  Simplify     \_a  p    x  a  p    >^  «^~"]  -p. 


158.  Simplify,  using  positive  exponents  to  express  the 
iswer,  fx~'^y^z~^\^      ( x'^X^ 

V   xy-^   )  ^\^y 

159.  Simplify     VT8 ;  36^  25"^;  ^^x-,  -\/W. 


r 

160.  Simplify      ^ 

161.  Simplify 


^h-- 


3^+1  gor+i 


-l^*+l 


162.  Simplify     5  ««  -  (5  a)0  -  l^  +  A. 

9^ 

163.  Simplify     5V|  +  V|-V8. 

164.  Simplify     3V12+V75-V108. 

165.  Simplify     ^^-^^. 


394  ELEMENTARY  ALGEBRA 

166.  Simplify     '^2-h-V2^2^. 

167.  Simplify      ^|  .  ^/IT, 

168.  Simplify     </2 -\- -^128  ^  <^, 

169.  Solve     Vrc  +  1  -  V2  a;  -  2  =  0. 

170.  Simplify     10Vi~3V||+7Vi-5V2  +  9VS. 

171.  Simplify      1 

V18+V27 

172.  Simplify     ■^_4V|  +  V^. 

173.  Simplify     V^-V^  +  #^^  +  2-x^!±Zr^. 

y         X  xy  xy 

174.  Simplify  ^  ~  ^^      - 

175.  Solve     Va^-8  +  a:-8  =  0. 


176.   Simplify     Vl8  +  \/-4-     "^    ' 


177.  Simplify     JV490  +  V|-VT60. 

178.  Multiply  V3  -  1  -  V5  by  V5  -  V3. 

179.  Simplify     (ViT^- Vr^O^CVrMH- VT^l). 

181.  Simplify,  using  positive  exponents  to  express  the 
answer,  r    « —  ,       ^    . 

182.  Find  the  numerical  value  of  the  following  fraction 

to  two  places  of  decimals :   ^""^^ . 

24-V3 


GENERAL  KEVTEW  W6 

183.  Divide  21  a^'' -^  27  ctf  -  26  a:2a  ^j.  20  by  5  -  3  ^^ 

184.  Simplify . 

r  4        1  1-3 

185.  Divide  J  -  h^  by  ■y/a--^'h. 

186.  Simplify     2  0:^9^1^81  +  27 Vi^TW. 

187.  Simplify     ^V^)!^5Zi. 

188.  Simplify      [a  +  5(1  +  V^^)]  [a  -  h(l  4.  V^^)] 
[a  -  6(1  -  V^^)]  [a  +  5(1  -  V:r2)] . 

189.  Divide  Va;^  by  Va;^. 

"\/2  -4-  2a/^ 

190.  Compute  the  value  of       _         _  to  two  decimal 

places.  V2-V12 

191.  Simplify   i  +  (i  +  ^+.v^x^^4-,^^r^-v^^Ti;^ 

192.    Find  the  square  root  of  a^  —  2h^a-{-%h  —  2h^a-^ 

193.  Simplify   ^yC^^-y-^r^-), 

y^  —  x^ 

194.  Simplify     pN^f. 

195.  Simplify       7-V2__^    T+V2  ^ 

V34-V5      V5-V3 

196.  Simplify      '^"^'^  1 


2-V3      2  +  V3 

197.    Simplify  ^ . 

Vir+1  +  V2:  —  1 


396  ELEMENTARY  ALGEBRA 

198.  Multiply  Jb'^  +  2  +  a~h^  by  a'h^  -  1  +  ah~K 

199.  Simplify     (V^  +  V^'' 

200.  Expand  by  the  binomial  theorem  and  express  each 
term  in  simplest  form     fa~^  —  —  ]  . 


201.    Simplify     V^  +  y  +  ^_V^+y-^ 
■y/x  -\-  y  —  z      ■\/x  -\-  y  -\-z 


202. 


203. 


Expand     («^  +  -Y- 

Simplify     fV3  +  V2Y     /V3W2y_,3^ 
VV3-V2/      VV3+V2/ 


204.  Write  the  sum   of   V28,    -V63,   and    V700   in 
the  simplest  form. 

/    I        i\_£iL       _i 

205.  Simplify     \^a^ -^  a^ j=^+i -f- a  v. 

206.  Simplify    8^-7V98+i\/i  +  -i:  +  ^-i." 

•^-  2^2      V2      3-2     30 

207.  Simplify     3*^+1 .  9^^~^  ^  2T  3  . 

208.  Simplify     g"*"! .  27"»+i  •  Sl"^  .  3-~-i. 

209.  Evaluate  a^Va^  +  3  « Va"^  +  2  aWa^  when  a  =■  4. 

Simplify  K-r>-yo(^^;-j;j^;)- 

211.    Simplify     ^-^-2  +  0.^" 


210 


212.  Divide     5x^-6x^-4:  x~^^  -  4  x~^  -  3  x^     by 
x^-2x-^. 

213.  Show  that  (29753)2-*.  43  -  (29581)2-4-  43  =  237336. 


GENERAL  REVIEW  397 

214.  Solve  5x^-Sx-2  =  0, 

215.  Solve  in  two  ways     5  a:^  +  14  a;  —  55  =  0. 

216.  Solve  8a;- 152:2-1  =  0. 

217.  Solve  ax^  —  hx=  c. 


218.  Solve  a;  +  5  +  2 V^+5  =  15. 

219.  Solve  by  factoring,  by  completing  the  square,  and 
by  f ormulge,  3  a;2  -  26  a:  +  35  =  0. 

22b.    Solve  4  a;2  -I-  8  mx  =  4  mn  4-  n?. 

221.  Solve  m^a^  —  mx  -\-l  =  x^. 

222.  Solve      .     ^J     ^=1         ^ 


t^^St  +  2  t-2 

223.  What  values  of  x  will  make  the  expression  (a:  +  2) 
(5  —  3  a:)  equal  to  six  times  the  value  of  a:? 

224.  Solve  and  check     -Vx  -f  2  —  Vx  —  6  =  Va;  —  3. 


225.  Solve     5a;2_32;_3V5a^-3a;-13  =  ll. 

226.  Solve     6(x^-h^-[-6fx-^-\-SS  =  0, 

227.  Solve     12fx^  +  ^-6fx-{--\- 291  =  0. 

228.  Solve  4a72  —  7a;  +  2  =  0;    give  both  roots  correct 
to  two  places  of  decimals. 

229.  Solve     (l  —  n^)x^—2mx  +  m^  =  0. 

230.  Solve     V^^^  +  x-S  =  0. 

231.  Solve     3VZ--|^  =  8. 


232.  Solve     V«^H-5  +  «^-l  =  0. 

233.  Complete  the  square  in  each  of  the  following  ex- 
pressions : 

a:2-12a;,  a;2  +  25,  a^  +  9a;,  a?^4--^j. 


:^I8  ELEMENTARY  ALGEBRA 

In  problems  234-7,  write  down  the  quadratic  equations 
whose  roots  are  given. 

234.  3  and  5. 

235.  1  H-  V2  and  1  ~  V2.  ^ 

236.  — -^ —  and  — -— . 

2  2 

237.  m-i-n-\-  Vw  —  n  and  m  +  n  —  Vm — n(m  >  n), 

238.  Solve  the  system 

fa;  +  V^+ «/  =  14, 
\x^+xy +  1/^  =  84:. 

239.  Solve  the  system 

f  a^-y^=^6, 

240.  Solve  the  system 

241.  Given  p  =  9A^^d;  find  value  of  d.  What  is 
the  positive  root  when  t=  .21  and  j?  =  3.2  ? 

242.  The  area  of  a  mat  of  uniform  width  about  a  picture 
10  inches  long  by  8  inches  wide  is  one  half  the  area  of  the 
picture.     What  are  the  outside  dimensions  of  the  mat  ? 

243.  Solve     -l-  +  _i- ^=0. 

x—a     x-\-  2a     2a 

244.  Solve    --±1-1=^. 

X—6         X-\-l         X  —  4: 

245.  A  loop  of  twine  30  inches  long  is  to  be  stretched 
over  four  pegs  so  as  to  form  a  rectangle  whose  area  shall 
be  50  square  inches.     What  are  the  sides  of  the  rectangle  ? 


INDEX 


[Eeferences  are  to  pages.] 


Abscissa,  252 

Absolute  value,  25 

Addition,  24 ;  associative  law  for, 
37  ;  by  counting,  30  ;  commuta- 
tive law  for,  34 ;  graphic  repre- 
sentation of,  30 ;  of  fractions, 
164  ;  of  monomials,  34,  35,  37  ; 
of  polynomials,  38  ;  of  surds,  279 

Algebra,  1 ;  laws  of  combination 
in,  9  ;  symbols  of,  2,  3 

Algebraic  expressions,  6 

Algebraic  fraction,  153 

Alternation,  242 

Antecedent,  239,  245   ' 

Approximations,  319 

Arithmetic,  the  notation  of,  1 

Associative  laws,  37,  55 

Assumptions,  16 

Axes  of  coordinates,  261 ;  of  refer- 
ence, 251 

Base  of  a  power,  11 

Binomial,  13  ;  cube  of,  115;  square 

of,  108 
Binomial  expansion,  300 
Binomial  formula,  300 
Binomials,  product  of  two,  114 
Braces,  12 
Brackets,  12 

Cancellation,  84 
Checking  the  result,  7 
Clearing  of  fractions,  191 
Coefficient,  10 
Common  difference,  365 
Common  ratio,  371 
Commutative  laws,  34,  54 
Completing  the  square,  316 
Consequent,  239 
Constant,  247 
Coordinates,  252 

Cubes,  sum  and  difference  of  two, 
132 


Decimal,  recurring,  379 

Degree  of  an  equation,  82,  142, 
143,  205,  312  ;  of  an  expression, 
100,  101  ;  of  a  monomial,  100 

Descartes,  261 

Difference,  27 

Distributive  law,  58 

Dividend,  33,  72 

Division,  33,  64  ;  by  zero,  16,  72, 
154 ;  of  fractions,  178  ;  of  mo- 
nomials, 66  ;  of  polynomials,  68, 
69  ;  of  surds,  281,  285  ;  rule  of 
signs  in,  33  ;  special  cases  of,  179 

Divisor,  33 

Equations,  15  ;  change  of  signs  in, 
84 ;  complete  quadratic,  313 ; 
conditional,  81  ;  degree  of,  82, 
142,  143,  205,  312  ;  dependent, 
207  ;  equivalent,  82,  208,  219  ; 
formation  of  quadratic,  327  ; 
fractional  and  literal,  191,  197, 
222  ;  graph  of  dependent,  260  ; 
graph  of  inconsistent,  260 ; 
graph  of  linear,  258,  260  ;  graphs 
of,  256  ;  homogeneous,  352  ;  in- 
complete quadratic,  313,  314 ; 
inconsistent,  207,  260  ;  independ- 
ent, 207  ;  indeterminate,  206  ; 
irrational,  321  ;  linear,  82,  205 ; 
number  of  roots  of  a  quadratic, 
343 ;  number  of  solutions  of, 
205,  208  ;  numerical,  197  ;  partic- 
ular systems  of  quadratic,  355  ; 
quadratic,  312  ;  quadratic  with 
complex  roots,  335  ;  rational  and 
integral,  142  ;  relations  between 
roots  and  coefficients  of  quad- 
ratic, 326;  roots  of,  82,  143, 
342  ;  satisfying  an,  82  ;  simple, 
82,  205  ;  simultaneous,  207,  229  ; 
solution  of,  82,   143,   206,  ^316, 


399 


400 


INDEX 


824  ;  standard  form  of  quadratic, 
812  ;  systems  of  quadratics,  350  ; 
systems  of  linear,  205,  208 

Elimination  by  addition  and  sub- 
traction, 209 ;  by  comparison, 
215 ;  by  substitution,  214,  350  ; 
by  undetermined  multiplier,  217 

Euclid,  57 

Euler,  109 

Evaluation  of  an  algebraic  expres- 
sion, 6 

Evolution,  302 

Exponents,    11  ;    fractional,    286, 

287,  288,  289  ;  negative,  286, 287, 

288,  289  ;  zero,  65 
Expressions,  algebraic,  6  ;  integral, 

100;    mixed,   170;  prime,    118; 
rational,  100 
Extremes,  241 

Factoring,  118 ;  equations  solved 
by,  142  ;  remainder  theorem  in, 
140 ;  special  methods  of,  134 ; 
summary  of,  136 

Factors,  9  ;  found  by  grouping 
terms,  121  ;  integral  algebraic, 
118 ;  monomial,  119 ;  of  alge- 
braic expressions,  118  ;  of  differ- 
ence of  two  squares,  126 ;  of 
general  quadratic  trinomial,  130  ; 
of  integral  expressions,  118  ;  of 
monomials,  119  ;  of  trinomials  of 
the  form  x^  +  ex  -\-  d,  127  ;  of  tri- 
nomial squares,  123  ;  sum  and, 
difference  of  two  cubes,  132 

Formula,  17 

Fourth  proportional,  245 

Fractional  equations,  191,  222 

Fractions,  153  ;  addition  and  sub- 
traction of,  164  ;  change  of  signs 
of  factors  in  terms  of,  156 ;  clear- 
ing an  equation  of,  191 ;  complex, 
183  ;  continued,  186 ;  division  of, 
178 ;  laws  governing  algebraic, 
153  ;  lowest  common  denominator 
of,  164  ;  multiplication  of,  161, 
178  ;  powers  of,  176  ;  proper  and 
improper,  170  ;  quotient  of  two, 
178  ;  reduction  of,  169  ;  reduction 
to  lowest  terms,  158 ;  simple, 
169  ;  signs  affecting,  154,  156. 

Function,  255  ;  graph  of,  256 


Gauss,  330 

Graphic  representation,  of  addition 
and  subtraction,  30  ;  of  the  rela- 
tion between  two  variables,  251 

Graphs,  251,  346 

Greatest  common  divisor,  146 

Highest  common  factor,  146  ;  of 
monomials,  147  ;  of  polynomials 
by  factoring,  148 

Identities  involving  roots,  267 

Identity,  81 

Imaginary  imit,  330 

Index,  11  ;  law,  55  ;   of  a  radical, 

275  ;  of  a  root,  265 
Involution,  800 

Least  common  multiple  in  arith- 
metic, 149 

Linear  equations,  82,  205 

Literal  equations,  197 

Lowest  common  denominator,  164 

Lowest  common  multiple,  149  ;  of 
monomials,  150  ;  of  polynomials 
by  factoring,  151 

Mean,  arithmetical, 368  ;  geometric, 
374  ;  proportional,  245,  374 

Means,  arithmetical,  368 ;  geo- 
metric, 374 ;  of  a  proportion, 
241 

Members  of  an  equation,  15 

Minuend,  27 

Monomial,  13 

Multiplicand,  31 

Multiplication,  81,  64;  associative 
law  for,  55 ;  combinations  of 
signs  in,  32  ;  commutative  law 
for,  54  ;  distributive  law  for,  58  ; 
of  a  polynomial,  58,  61  ;  of  frac- 
tions, 161,  173  ;  of  surd  expres- 
sions, 281,  282  ;  rules  of  signs  in, 
32 

Multiplier,  81 

Newton,  801 

Notation,  81,  106 

Number,  complex,  333  ;  imaginary, 
880  ;  irrational,  262  ;  literal,  1  ; 
prime,  118 ;  rational,  262 ;  re- 
ciprocal of  a,  178  ;  symbols  of,  2 


INDEX 


401 


Numbers,  algebraic,  22  ;  conjugate 
complex,  333  ;  negative,  22,  23  ; 
opposite,  25  ;  positive,  22  ;  prime 
to  each  other,  158;  property  of 
positive,  266  ;  real,  330  ;  scale  of 
positive  and  negative,  23  ;  sum 
of,  25  ;  use  of  Uteral,  1 

Operations,  order  of,  7  ;  rational, 

100  ;  symbols,  2 
Ordinate,  252 
Origin,  251 

Parentheses,  12,  48 

Pascal,  204 

Plotting,  253 

Polynomial,  14 

Portraits,  Descartes,  261  ;  Euclid, 
57  ;  Euler,  109 ;  Gauss,  330 ; 
Newton,  301  ;  Pascal,  204  ;  Py- 
thagoras, ii ;  Vieta,  33 

Powers,  11  ;  ascending  and  descend- 
ing, 61 ;  fundamental  identities 
involving,  263  ;  of  a  fraction,  176 ; 
of  i,  331 ;  quotient  of  two  of  the 
'same  base,  64 

Processes,  fundamental,  34 

Product,  31  ;  of  conjugate  surds, 
284  ;  of  two  binomials  having  a 
common  term,  114  ;  of  two  com- 
plex numbers,  334  ;  of  two  con- 
jugate complex  numbers,  333 ; 
of  two  fractions,  173 ;  of  two 
monomials,  56  ;  of  sum.  and  dif- 
ference of  two  numbers,  112 

Products,  type,  107 

Progression,  arithmetical,  365  ;  de- 
creasing, 372  ;  geometric,  371  ; 
increasing,  371  ;  infinite  geo- 
metric, 377 

Proportion,  240 ;  by  alternation, 
242 ;  by  composition,  243 ;  by 
division,  243  ;  by  inversion,  241  ; 
continued,  244 

Proportional,  fourth,  245 ;  mean, 
245 ;  third,  245 

Pythagoras,  ii 

Quadratic  equations,  312 ;  com- 
plete, 313 ;  incomplete,  313 ; 
number  of  roots  of,  343  ;  particu- 
lar  systems   of,   355 ;    standard 


form  of,  312  ;  systems  of,  350 ; 

with  complex  roots,  335 
Quadratic  surd,  266,  295 
Quotient,  33,  72  ;  of  two  fractions, 

178.  • 

Radical,  265 

Radicals,  275 

Radicand,  265 

Ratio,  239 

Ratios,  composition  of,  245 

Remainder,  72 

Remainder  theorem,  139 

Review,  52,  75,  137,  186,  234,  296, 
381 

Root  of  an  equation,  82 

Roots,  265 ;  approximate  square, 
308  ;  cube  root  of  a  monomial, 
105 ;  found  by  factoring,  143 ; 
like  and  unlike,  265  ;  nature  of 
the  roots  of  ax"^  -^  bx  -j-  c  =  0, 
340  ;  principal,  265  ;  quadratic 
equations  with  complex,  335 ; 
square  root  of  arithmetical  num- 
bers, 307  ;  square  root  of  a  bi- 
nomial surd  expression,  293 ; 
square  root  of  a  monomial,  105  ; 
square  root  of  a  polynomial,  304  ; 
square  root  of  a  trinomial,  303 

Rule  of  signs,  in  division,  33  ;  in 
multiplication,  32 ;  in  subtrac- 
tion, 28 

Series,  365  ;  sum  of  arithmetical, 
369  ;  sum  of  geometric,  375 

Signs,  affecting  a  fraction,  154 ; 
change  of,  in  terms  of  a  fraction, 
156  ;  like  and  unlike,  22  ;  of  ag- 
gregation, 12  ;  rule  of,  28,  32,  33 

Square  of  a  binomial,  108  ;  of  a 
monomial,  102  ;  of  a  polynomial, 
111  ;  of  a  trinomial,  110 

Subtraction,  26  ;  of  fractions,  164  ; 
of  monomials,  42 ;  of  polyno- 
mials, 44  ;  of  surds,  279 

Subtrahend,  27 

Surds,  266 ;  addition  and  subtrac- 
tion of,  279  ;  comparison  of,  278  ; 
conjugate,  283  ;  division  by  poly- 
nomial containing,  285  ;  multipli- 
cation and  division  of,  281  ;  order 
of  a,  266  ;  product  of  conjugate, 


m2 


INDEX 


'284  ;  qnadratic,  266,  295  ;  simpli- 
fication of  fractional,  276  ;  square 
root  of  a  binomial   expression, 

Symbols,    2 ;     of    number,    2 ;   of 
operation,  2  ;  of  relation,  3 

Terms,  13  ;  like,  14  ;  6f  a  fraction, 

163  ;  of  a  proportion,  240 
Transposition,  83 
Trinomial,  14 


Type  forms,  11© 

Value,  arithmetical,  25  ;  numerical, 
25 


Variable,  247 
Variation,   247 

verse,  248 
Vieta,  33 
Vinculum,  12 

Zero,  23,  25 


direct,   247 ;   in- 


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